Transcript for:
Understanding Vector Multiplication and Dot Product

all right welcome back um today as I I just mentioned introducing it to you guys we're going to answer the first part as a two-part answer to a oneart question we know how to add vectors we know how to subtract them can we multiply vectors and the answer is a two-part yes but there there's two different ways to do it uh and the first way that we can consider this idea of of a product amongst vectors something called The Dot product this is what right now the dot product you're not going to use it for anything right away you're like oh yeah that's what that's what it is like when you add vectors you get this other Vector that's not going to happen right now uh what's going to happen is I'm going to show you the operation we're going to treat it like an operation I'm going to show you like 10 different ways we use it uh so the dot PL is in lots of stuff it doesn't specifically give you anything right now that you're going to completely understand right away don't worry about that I just want you focus on operation and then through the course of this lesson I'm going to show you a lot lot of ways that it's used you guys get the idea so first thing what the dot product does it is a way to multiply uh but it does something kind of special what it does it takes the components of the of the vectors our position vectors it multiplies corresponding components and then adds that all up so it adds the product of corresponding components that's what a DOT product does so what this does write this down is a DOT product is going to add the products add the products of corresponding components of your vectors you go oh right what's what's that me because if you never seen it like I don't even know what you're freaking talking about right now let me show you it's it's actually really really really really easy to do but you got to do it at least a couple times to get that the hang of it so let's imagine that I've got a couple vectors I've got a and I'm going to use this Vector bracket notation and B leave this blank for a second we're going to we're going to discover what that actually does with our first first example it's going to be interesting it's like what uh and we we'll talk about that so let's look at what this would do so it says you're going to be adding some stuff together but the stuff you're adding is a product of corresponding components that means pieces that match up so like the X components multiply those the Y components multiply those the Z components multiply those and then when you have multiply those add them all up so what this would do with these with these two it would do A1 time B1 plus A2 that's that's our y components here A2 * B2 + A3 * B3 you guys see what I'm talking about the the products of corresponding components you guys see that say take your X's multiply them take your y's multiply them take your z's multiply them these are inner vectors and what and when you add them what this gives you is a dot b if you are talking about vectors and you put a little dot in there that's one of our signs for multiplication right it's one of our answers here uh that's called a DOT product so that the dot between vectors means a DOT product head not if you're okay with that so far now this is what's a little weird about it check this out Vector right yes okay Vector yes Vector no no no component yes yes Vector no component Vector no Vector yes no scaler weird this dot product operation is a way that we multiply our vectors and it actually gives us a scale this is going to give you a number oh that's all I can do look at that you're multiplying numbers you're multiplying numbers you're multiplying numbers and you're adding numbers that's all you're doing it's not a vector this is equal to c a scaler so what I had you leave blank up here this this adds the products of corresponding components of two vectors and it gives you a scaler how we use that scaler oh my gosh it's used in in every it's used in a lot of stuff that we're going to be doing okay we always use it it's not hard man it's really easy but you need to know that we're not getting a vector out of it that that's why it doesn't make a whole lot of sense right now and say hey let's multiply and what's it give you ah it's a number it's not a vector it's a way that these vectors relate to one another yeah and we're going to talk a lot about that but for right now the thing I want you most to understand is that we get a scaler out of this dot product operation show hands if you do understand that that's fantastic let's practice a couple just one right now to make sure that we got this and then we're going to continue actually you know what do you think you can do it can you do on your own right now can you find the dot product of V and W can you do that let's try it's kind of one of the easier things that we get to do in this class oh man we're multiplying stuff and adding it that's that's awesome as you're doing it I want you to think about this I want you to think if it would matter the order in which you did a dot don't answer it now just do it uh do do this figure out the dot product of B andw make sure you're getting a scaler out of it and then we're going to talk about some of those properties can you reverse them can you distribute them can we do things like this for have I given you enough time to at least do the dot product of that thing let's let's see how it works um honestly there there's not a whole lot of work I need you to show here frankly I don't care um me personally I show it because it's really easy for me to make mistakes in my head like negative positive sort of thing so I show I at least show the three numbers before I start adding them together does that make sense I'm going to show you everything for the first two but after that it's it's kind of up to you so here's what the doc product does it says number one thing you got to have two vectors for a DOT product to make sense and you got to be able to identify the X components and multiply them the Y components and multiply them and the Z components and multiply them so let's look at the X's first what are my two x components one from each Vector what is it good does that negative matter it doesn't matter when we doing magnitude because we're squaring stuff but we're not doing that now so the negative matters so this is X component X component product of corresponding components we got that then we're gonna add some stuff now we look at the Y component come on everybody kind of quickly y components what are they y component include the sign of course we got3 y component we got two and then we're going to add together the product of the Z components what's the z component here good okay and yeah that's pretty pretty much all the that that is all the dot product does go okay well we got2 we got -6 we got how much is that going to give us how much does a negative matter yeah it does matter it's not not a magnitude here it matters so the dotproduct of V * W is- 10 it's not a vector it's a scaler what's that scaler mean I don't know right now I don't know okay it's it's just not operation as we move forward we're definitely going to know what this thing means okay but right now it's can you do the dot product show hands if you can easy medium Hardy it's really easy I mean you're multiplying numbers and adding them and that's that's as hard as it gets right because it's a 3D Vector we don't have 40 vectors so that's fantastic so literally all you got to do is multiply three three sets of numbers and and add them up that's it you with me now the properties as you working through it does it matter if I reversed these two numbers and reverse these two numbers and reverse these two numbers which means that the dot product has commutativity it doesn't matter the do the way that you uh multiply those numbers as commutative and since this is multiplication and addition those are both commutative actions or operations this is also commutative and there's some other ones if we have an addition of two vectors and we dot product that it's also distributive if we have a scaler so scaler times a vector and then we dot product with another Vector it's reass it's associative which means I can move that scaler wherever I want to because I'm just multiplying through anyway uh I can do this I can do U this one so I can do the dot product first first and then multiply with the scaler or I could do scaler times a second vector and Dot product what I'm saying is it doesn't really matter wherever that scaler is at you're going to get the same answer real quick if you're okay with those ones they're kind of intuitive it's like oh yeah this stuff same sort of stuff we have with most of our our math this is a little weird it's the first time I've introduced it here that little zero with a vector on it means is the zero Vector if you don't know what that is it's kind of basic but it's the vector that doesn't go anywhere it doesn't Point anywhere it's still a position Vector but it's this position Vector it's 00 0 so it's it's at the origin it literally is the origin considered as a vector guys are you with me it's just a point yeah consider a vector we can use it as a vector yeah and otherwise if we didn't have a vector I couldn't talk about the dot product right now does that make sense so the zero the zero Vector times any Vector it gives the zero Vector back again I mean multiplying by zero right and that's basically what we're doing so we go hey multiply everything by zero you're going to get zero you're going to get the zero Vector oh I'm sorry um yeah I made a mistake you're going to get zero so when you add them together you're going to get zero yeah erase that Vector because dot products always give us a scaler now the last one the last one's man it's really interesting what happened happens when you dot we're going to spend some time on this okay what happens when you dotproduct a vector with itself I want to go through the work and kind of prove this out because we get a couple results that are that are interesting that I get to use later are you guys ready for it so what what would a a vector dot producted with itself look like let's consider this to be that V1 V2 V3 and then we're going to do product that with another V1 V2 V3 it's going to be the same exact components you follow so that would look like this oops pretty straightforward hey if I asked you to um could you do the dot product on these these vectors right now if I ask you to okay quickly what's the first two things you're multiplying together what is it just like we did plus what I'm guessing you know the last one true or false that's going to give me a vector true or false what's it going to give me just like every other time do prodct Workforce you're going to give us a scaler that's right but wait a second wait there's more how else could you represent B1 * V1 awesome but there's more than that in fact what this man this looks real if you okay if I asked you to find the magnitude of V1 could you do it explain to me come on quickly how you find the magnitude of V1 what do you what do you do square root okay you have a square root right and then inside that inside the square root what would be thereon X component squ y plus the Y component squar plus the Z component squar exactly this piece would be inside of the magnitude does that make sense this right here is the square of the magnitude that's exactly what it is in fact you I'm to that undoes whatever I'm doing right so square root squar gone it's all positive so I don't have to worry about negatives because I'm taking these numbers and squaring them and adding them it's positive now that one I mean you got to be okay with it but that that is the magnitude of V1 are you following what I'm talking about that's it this is the magnitude of V1 squar so this is the magnitude of V1 squ which means we got a we got a couple things I'm going to write up there I'm going to kind of move up to the top of the board a couple things I want you to know some some corollary some really interesting stuff that we're going to use right now it's like okay that's awesome man what are we going to use it for I'm going to show you as we going through the Section okay right now we're just kind of building properties CU it's probably the first time you've ever seen the dot product unless you've had the class before or like a physics class where maybe you had to do it you're like what's what's it all mean I don't know you just do it uh so we we're figuring out what it means now what we can do with it did you follow this hey do prod the same Vector you get have the same components obviously it's going to be squared that that looks great uh but but wait that's the inside of a magnitude which means it's the magnitude square is what that is and that's one of the three pieces we're going to write right now so here's how we can think of little subsection so five continued if you will the first thing we can think of is if you want to dot product a a vector with itself it's it's pretty easy to do because literally all you're doing and you're like well this doesn't look easier uh it's finding the magnitude and squaring it but what that means is that all you have to do is take one it's common sense I mean it's common sense right here you're just squaring the the components so if you ever wanted a dotproduct of vector with itself you literally just have to do B1 2 v2^2 and B3 2 but we use these facts in some proofs later on we're going to we're going to be using that stop so it's it's it's valid for us it also means this if you wanted to do magnitude another way and you've already calculated the dot product which I don't know why you would but you can solve for that so it's a different way to think about magnitude so there's some some kind of interesting little little tools that we have up here I want to know if these make sense to you this one probably should be the most obvious one okay it says sayy yeah same component you're squaring that's all you're doing but I need you to understand that when you do that what you're getting is the magnitude squared you're eliminating the square root you're just you're squaring it it's no square root that that's it that's what I want you to get show fans if you do you feel okay with it all right let's move on let's go ahead and we're going to try um three yeah three examples with the following vectors this is purely for practice I just need to make sure you're you're getting this down okay so I'm going to do maybe the first one for you and the last one I want you do the middle one on your own so that's that's what's going to happen do you feel pretty comfortable on finding the dot product I want to ask you to okay you know some some people at this point uh get a little confused about the the letter U am I talking about a unit Vector right now when I put the U up there how would I talk about a unit Vector if I wanted to do that a little half that's what would be there so I'm using a lot of different letters so you get used to it I don't want to just always use a b and c or B and W because a lot of textbooks out there use a lot of different letters so I'm going to use a variety of them just to keep it fresh um but I don't want you get confused like oh every time Leonard says you it's a unit Vector that's not true every time we put that little hat up there that's a unit Vector that's what that means like that that's a unit Vector in the X Direction all right hey practice time here's what we're going to do uh the first thing I want you to do is we'll do this first one together so you get a hang of it if I asked you to calculate this V do product with W + U could we do it yeah of course going be possible well maybe that's what I want you to think about actually first is is is it possible first and second what are you g to end with are you gon to end with a scaler are you going to end with a vector can you even do the work so let's try it what's going to happen when you add together W + U are you going to end with a vector or a scaler very good can I dotproduct a vector with a vector so that's possible what am I going to end with SC beautiful that's exactly what I want you to think about because sometimes you know book's tricky they put one up oh not possible oh crap I'm trying to dot product a scaler with a vector and you can't do it uh that doesn't make sense so let's go ahead and try this I I don't really care how much work you're showing or not showing show enough so that you don't make the silly mistakes you know what I'm talking about sign errors things like that me personally I would do it this way I'd write out this Vector I'm not trying to dot product that right now I want to make sure I know how to add vectors those two vectors first so I'm hanging on to this one I need to translate one of these I don't care which one but I'm going to always give this to you so you practice that translation since that's already in that vector format with those brackets I'm going to be trying to do that and then I'm going to be adding W and U together so I'm looking X component I'm looking X component and I'm adding it how much do I get with that do I need to write the zero in this format do I need to write the zero in this format if I had I's and JS and KS would I need to write it okay cool so I need to write the zero here how about um oh wow how about the next one what do I get Z I give you a really funny example how with the next one what do you get two you're adding that's one of the little mistakes right be careful be careful on this stuff won't that suck when you get to your first calculus 3 test and you know how to do all of this and you go oh I get zero when I should have got two are you going to be very happy when I hand that back to you no no you're going to you're going to be so mad at me I'm like hey dude or if you're a girl like hey chick no I'm going to say that but no you're not going to be happy be care please be careful even if you have to bust out your calculator and do it Problem by problem I don't care uh make sure you get the right thing better get the right answer slow than the wrong answer fast right take your time not too much time but but take your time well now it makes sense I got a vector dotted with a vector let's go ahead let's let's finish this off dot product does one thing and one thing only it takes corresponding piece is so it's nice to have the same format it multiplies them and adds them did it give us what we expected did we get a scaler like we thought we would get yes that's what I want think about what you're getting first then make sure you're getting that thing can you try one on your own here real quick yes okay the answer that question by the way is always um always yes so here's my question I want you to think about this before you answer it maybe just hold it inside for a little while here's what I'm going to ask you don't answer right now is that a DOT product is that a DOT product and then what are you going to end with think about that before you start and then do the problem okay don't you don't need to answer out loud just keep it up here if you finish that one really quickly I'll put our our last example over here and you can try work on that on your own as well I know you're still working on it but I wanted you to think about this here's my question first question is that a DOT product right there no why not what's that give you scale scaler it's just a scaler times a vector is that relevant can you do a scaler time a vector what are you going to get out of it you need to be ending with a vector here does that make sense you get it that's fantastic now see what it is first thing I'm doing is doing the dot product the dot product here gives us and you can do it like this if you'd like you can do -2 + -12 + 2 if you want to write like that that's fine that's exactly what we're going to get multiply this -2 multiply this2 multiply this two just don't make a sign error you not do that and then we go okay after we find that scalar just that number then we're going to be multiplying it by this Vector not too bad but you you definitely have to understand all the operations involved show hands you made it as far you feel okay with that one how much does all that work out to 12 yeah2 two gone so we got we have that could you keep going could you stop there uh not really in this case when you have unit vectors yeah you can stop with that uh when when I ask you to do a full operation I want you to do it so let's go ahead and let's continue here we would distribute this scaler through our vector here 48 - 12 and hopefully I got it right did you guys get the same thing I got yeah sounds good are you sure you're okay with it do fall under like easy medium hard for you what what do you think not too bad you but you got to understand how to do it and don't make the signers oh my gosh that's what causes your teacher to drink at night do you know that Cal 3 oh my gosh yes and doing the integral of cosine and giving me negative freaking sign oh if you do that I swear my wife is not like I hate Li my students are killing me you're literally killing me literally it breaks my heart just a little bit every single time like no the integral of cosine is positive sign not negative sign plus C you freaking kids you're killing you're kids all right uh but it kills me so don't do that and don't make sign herge uh now the last one did you start working on that I was was rambling off and lamenting about my poor sad life when I go home it's not sad it's awesome but uh anyway I didn't tell you I won that $1.4 billion on the lottery now you're really thinking aren't you I didn't I wouldn't be here I I tried to lie to myself like yeah I'd still go I'd still go like no I wouldn't dude there's no way iing do that uh any I might I don't know no I'm Dreaming sorry I didn't win the loty let's continue math cuz I have to make a living somehow so um did you start it that's fine if you didn't we do it right now I wish you would have though would have made me look good on camera thanks a lot people um so what are you going to get though what's this going to give you scalar Vector what's just this inside piece V Vector good darn thing because you can't take a magnitude of a non Vector all right it doesn't make sense that would be called absolute value so we have vector magnitude gives us scalar squared SC plus the same exact thing but you're adding so another scaler we're going to end with a scaler here I'm going to go quite quickly through it see if you can hang on I want your participation though if we do V minus W that's that's just these two what's the resultant Vector that I'm going to get can you tell me what that resultant Vector is going to be I heard the three what else we're subtracting so subtract component for component that needs to be this Vector right there does that make sense now let's see if you're paying attention a little bit earlier I'm going to work this side down then I'm going work this side down we're just going to put them together okay that's how I would do the problem so let's let's go all the way down with this thing if you were paying attention earlier and you kind of you kind of saw this how what's an easy way to do the magnitude of a vector squared tell me how you would do that square each of them would you take a square root if I want this it's the same as this just Square them all that's all this is going to be does that make sense otherwise you're going do the square root and they go oh and then I got a square that's a waste of time so this says hey let's do 9 + 49 + 1 3^ 2 9 7 2 49 1 2 1 explain to me in in your own words why I don't have a square root around it even though it's a magnitude why don't I have that very good yeah the square here says uh magnitude square root square gone it's it's this idea how much and then we're going to add something to it have you already added up the V plus the W have you done that right side tell me what it is what you get are are you finding what I'm talking about about all sign errors have you have you done some of those yet I do them all the time I try to be careful but it happens that's why you got to double check your work on these things even though it's fairly easy straightforward I mean you're doing like at most pre-algebra level stuff uh but it's very easy to go oh I'm working with the wrong vector or oh I just subtracted I should have added something like that okay it's really easy to do that so I'm going to I'm going to double check right now so let's make sure we're right -4 plus is that right oh no I the wrong one so 4 +3 I got that one looks like we're good how do I get this again one how much do I got there 11 total of that's what I want to see two P feel okay with the idea any comments questions anything before we continue because it's going to get real after this you're going to find the first thing out uh about dot product and and why we use it so anything do you feel okay on doing Dot product and understanding whether we're going to get a vector or scaler when we're doing this stuff yes no having fun yet you can live say yeah rather be no other place I I don't want to win the lottery cuz I learn this right you know you're super math dork when you get excited that the Whiteboard erases cleanly this one you know I made it I finally made it yay um do you know what the law of cosign says random question he's like how's that relate to anything we're doing very strongly do you know what the law of cosiness says you do I don't I have to look it up on my notes here real quick but I'm going to write down so law coides well cosine says if you have this triangle any any triangle where the angles of this triangle are given by these capital letters a b and c and the lengths of the side the keep this in mind please keep this in mind the lengths of the sides are given by little a little B and little C so stand for the lengths the lengths of the sides what the law of cosine says is a way to relate these angles to the lengths of the sides it says this just so you know those are three numbers in there say all right hey take this triangle what I know is that the length of a s little a length of a s equals this S Plus this s - 2 * this length time that length time the cosine of the angle between those lengths look at the relationships of what's going on okay please understand something this is the biggest thing I want you to know this angle is between same side right between those two sides that's what I want you to know do do you see what I want you to know yes no and it has to do with the length of the side opposite that angle that's what I want you do you get what I'm I'm saying here now why are we doing this we just switched to trigonometry we did but but maybe we think about it this way I'm not going to use abs and C's because we have a lot of those on the board right now but what if we have two ve two vectors oh wait a minute dot product works for two vectors that's interesting uh and we're trying to talk about this angle they obviously have an angle between them and this is in three space okay so this doesn't have to be on a plane this is just two random vectors if they are not parallel they are going to meet somewhere and they are going well if we consider them to be position vectors they are going to meet at the origin and they're going to have an angle between those two things you understand what I'm talking about well let's let's let's do something uh the first thing I want to do is build a triangle out of this oh man are any of you guys good at all that stuff we practice from 11.1 you know what that is yeah that's um that's uh isal to nope just this this vector what what's that Vector huh yeah I know it's a resultant Vector I'm asking what Vector it is how would you relate it to V and W is it V plus W no if anything this Vector plus whatever Vector equals this Vector that's how vector addition works well you could okay you could do that call this uh I don't know then W + I don't know Vector equals V it's just symbols man it's just symbols and if you solve for this then I don't know Vector equals vus W solve for it it's vector addition all those properties actually do work then question mark Vector is the resultant Vector from that is vus W click show hands feel okay with with that idea did you know you could do that it's pretty cool that's because they're like touching from head right initial terminal terminal initial V goes from initial to main terminal and we can do vector addition subtraction just like that now here's what I'm going to do what I'm going to try to do is fit this stuff on that picture I've tried to draw exactly the same for you so let's let's first look at it can you tell me oh and be careful with it what's taking the place of a what is it okay hang on one person said it a lot of you said V minus W and that that's not right and I said for reason for a reason think about it I said this a is what I said it you should memorize every word that I've said uh this is the what's the A stand for come on what's it length length length is this a length no how do you find the length of a vector oh crap okay so when I'm talking about a and I put it on this this picture right here the thing that's corresponding with a is the magnitude of vus w^2 oh my gosh we just talked about all that right now what what's equal to let's fill it out what's B okay tell me what B is first what's it represent the little B what's it what's it represent it represents a length that's right what Vector is overlaid on top of B which one so we're talking about this how can I represent this idea of a length of the side given that that's W come on quickly what is it all I heard but I'm imagining you said magnitude the magnitude of v² huh let's see if you can follow it I want to make sure we we've talked about it just right now but I want to make sure do you understand that a s the length of this is the magnitude of vus W2 heading on if you do I need to get through it do you understand that the W part the B is the W part it's the magnet this is the length the length of w ^2 the length of v^ 2 minus this is a weird part it's 2 * the length of W Times the length of V time the cosine this is how I want you think about it not cosine OFA whatever Theta is it's cosine of this please listen time cosine of the angle between whatever those two vectors are this a is the angle between these two sides this Theta is the angle between these two vectors do you guys get what I'm talking about that's important so Theta yeah I know it's the only angle that I've listed but it's the angle because it's between these two sides it has v and W adjacent adjacent vectors you get what I'm talking about we're gonna do a whole lot of work so first thing keep this all in your head we're going to work just on this side we're going to do a lot of the stuff that we just talked about number one uh the the reason why that why we actually did this was for this I now have a magnitude squared and what I know from this and this is the this is the big deal okay here's like the punch line of this are you ready you're not going be ready he like sh what uh this is where the dot product comes in okay right here do you see how I have a magnitude squared a magnitude squared is a DOT product it is so this side what this gives us is vus W dotted with vus W equal to the same bunch of crap that I'm just going to have up there because I'm lazy and I don't want how to write the whole thing make sense Have I Lost You Yet you guys haven't have fish eyes out there are you guys okay yeah this is the fun part come on we actually get to do something don't give up on the easy stuff like oh I just want to add stuff I can do I can do dorky voice to you too can we do that let's do dorky voice teach the whole class just like this all my subscribers gone sorry guys uh sh I do dorky voice but you got to get what I'm talking about a magnitude squared is a DOT product of this the inside thing dotted with the inside thing do you guys get it but but wait we just learned out that we learned that dot product is commutative and it's associative it's it's all these things that we normally have which means that we can do this equals same bunch of of of junk guys are you okay with that I'm literally just Distributing can you guys do that okay here here's the the awesome part about it you know how we did this right we said hey magnitude squared is a dotproduct of the same Vector dotted with itself correct we can go backwards if this is V Dov v.v is the magnitude of v s are you paying attention we're proving something pretty substantial here that we're going to use for a lot of this at least this chapter this I want you to hang on to keep it right there this same thing magnitude of w^2 did you did you write it down I don't want to lose you you got it equals all of this stuff and then the magic happens see I'm going to write it in purple so you guys see the trans transition here so the stuff in purple was a stuff we originally had right here now I want you to take a close look at it can you see some pretty cool stuff that's can happen it's can cancellations so that means when we subtract hey can we subtract let's think about it uh vector or scaler come on everybody quickly Vector scaler you can subtract SC you're subtracting numbers that's it that's all it is it's a number Gone Gone cool anything else I see more Gone Gone yes yes no now we're we're all adult mathem mathematicians right so we can do a couple things at once in our heads we just eliminated this and this and this and this we have one term equals one term they both have negative - tws on them I know I'm not going to add -2 but I could divide by -2 do you see it so when I when I do this do you need to see that step do you want me write that for you yeah okay I'm also going to switch these around they are scalers that means they are numbers which means we have commutativity which means I can just switch them why because I wanted to match up and look nice now you see it go okay hey let's uh let's divide by -2 then this is gone and here's what we end up with number one we end up with this was pretty cool che check this you don't even think it's cool yet you will you end up with a different way to do dot product so say hey if you want to do the dot product yeah it's pretty easy to do but let's say you didn't even have the vectors themselves let's say you just had the lengths of the vectors and you had the angle between them could you do the dot product with the lengths of the vectors and the angle between them yeah length time length time the cosine of the angle between them that's a different way to do the dot product that's pretty cool here's the other one the main core so that's number one okay I I'll list that out in a little bit also here's the main one can you solve for cosine thing that's one of the big reasons we need the dot product why what's it do nothing I don't know don't pee your pants it's okay it's fine that's what it does it gives you the angle between any two vectors that you that you want so let's say I gave you V and W or a and b or any two vectors in the world could you find the angle between those two vectors absolutely that's pretty easy you know how to find what's that me magnitude scaler right magnitude this one's a magnitude do you know how to find magnitudes you better at this point do you know how to find a DOT product yes scaler scaler scaler scaler scaler can you figure out an angle if you have cosine Theta equal to a number yeah all you got to do is hey cosine inverse of whatever this is done you just found the angle between two vectors this is how you find the angle between any two vectors it's pretty cool the reason why I wanted you to notice what the Theta was is because Theta is the one that we Define to be between V and W so when we did all this work that's the angle we're finding no other stuff it's the angle between any two vectors that you do product and divide by the mag which is neat it's pretty cool couple notes actually a lot of notes that that I want to do here here uh the first note you need to make a note that this works for Theta between 0 and Pi so it's not 0 and 2 pi and here's the reason why if it was if you have an angle think about this for a second if you have two vectors right and your angle is this is my fifth favorite dance move behind uh the one /x you know that one it's but it's this one your angle moves okay so if you have this angle that's more than Pi you able to go this way and get a smaller angle does that make sense so we Max it out at Pi because anything further than that it's a smaller angle the different way which is what it's going to give you so that's number one now how about this what if Theta is zero or if Theta is pi just think about it logically for me okay think about if you have two vectors and the angle is zero between them tell me something about those vectors oh wait same Vector what if they're different lengths are they the same vector no same I love how one of you said same unit Vector what's the same unit Vector mean yeah you're right same direction what happens if two vectors are going the same direction what are they they're parall so if our angle is zero we have parallel vectors what if our angle is pi do we also have parallel vectors yes yes opposite direction but they are parallel does that make sense so if this happens if Theta is z or Theta is pi we have parallel vectors what about this one and we're going to work with this one this is going to be kind of cool come on think about what we just did okay what if the angle between two vectors is exactly Pi / 2 come on what is it if the angle's zero they're going the same direction if the angle's Pi they're going opposite directions but we talked about having the same unit Vector right which means parallel they're scalar multiples of each other just a negative that means parallel vectors are scalar multiples are parallel that's important what if the angle is pi/ 2 what do you know about those vectors perpendicular we have a different word in Cal 3 we throw a lot of words around that mean the same thing one of is perpendicular one of them this means the same thing is called orthogonal can you say orthogonal orthogonal yeah it's a weird word another one's called normal normal means perpendicular also orthogonal and normal when we use those it has a connotation that we're in Three space not just like perpendicular on a plane okay so that's why we use them uh but it's it's no different so if if we have th / 2 we got per pendicular vectors we call those things orthogonal or normal we also talked about this I move over here a little bit uh before I do that I want to do like a 30 second recap just to make sure that man we just did a ton of stuff all right and we proved it which is pretty cool I probably could have just given this to you but you'd be like what did you follow that proof oh crap don't make me redo it oh my goodness please I'm begging you uh do you know how to find the dot product between two vectors do you know how to find the magnitude of any Vector I tell you pretty easy stuff now using this that we just proved it we now know how to find the angle it's based on the law of cosines uh we just took some lengths of vectors those are called magnitudes we put them in the law of cosin we solve for the cosine of the angle between two vectors that that's what we did if you can find the dot product of magnitudes you can always find the angle between two vectors using just a cosine inverse have I explained that well enough for you guys understand that okay after that we go okay well what what happens what are the what are the cories here I mean what what's the purposes well we know a couple other things if the angle between two vectors is zero you they're in the same direction they have the same unit Vector if it's Pi they're a scalar multiple just a negative of each other maybe more or less uh but that that also means that they're parallel if we have pi/ 2 we have this 90° thing that means orthogonal these perpendicular vectors and now I'm going to use this the alternate form of the dot product to to show you something that that's pretty neat it's it's um well you'll see so let's use the alter form of the are you still are you focused in right now have I just kind of beat your heads into submission just a little bit or you going all I heard was normal and you didn't even hear abnormal cuz I never said it but that's okay I don't think I did I don't have to watch that video I don't think I said abnormal that's something psychology I'd already told you they won't give me a site degree I'd be too dangerous yeah wish be awesome I wish I was a site no I don't I got too many problems deal with other people uh so you believe that right because it was like halfway through our proof I proved it it it was there now let's take this and run with it if two vectors are perpendicular orthogonal normal the angle between them will be pi/ 2 are you are you with me yes so four perpendicular vectors for orthogonal vectors Thal / 2 notd your head that you that you heard but you understand the concept let's plug it in let's see what Happ let's assume that V and W are perpendicular or orthogonal okay so if V and W are orthogonal then the angle between them is pi/ 2 now your head that you're you're with me let do it so then V do W would equal okay whatever this is times whatever the magnitude of w is time cosine but wait if you're assuming that V and W are perpendicular and we know that the angle between them is Pi / 2 by the way verify this you can't have 3 pi over 2 you can't can't because if you went past Pi you could have measured it the other way make sense we don't care about 3 Pi 2 we care about pi over two 3 pi over two is perpendicular it's orthogonal you follow now you tell me right now what's cosine of pi 2 what's a scaler time a scalar time a zero I'm getting the chills right now it's so cool this is how you show that two vectors are perpendicular if you do a DOT product and you get zero the only way that happens look the only way that's a magnitude right if you have a nonzero Vector so it actually has a length that cannot be zero unless you have the zero Vector if you have a magnitude it's nonzero length it's not the zero Vector so given two actual vectors the only way this ever happens is if you have pi/ 2 everything else cosine has a value for does that makes sense and you can't have 3 two because other way so this right here this so if I ask you on a test how do you show that these two vectors are perpendicular how you going to do it dot product if the dot product is zero are your vectors perpendicular if the dot product is not zero are your vector perpendicular that's it this this is how you show two vectors are I'm going to use the fancy word all right orthog or perpendicular the same thing head not if you're still okay what we're talking about um couple other things I want you to think about this for a second you know what I J and K are correct yes tell me the relationship between I and J and I and k i and perpendicular how about I in K you can just nothing I'm not going to get nothing out of you guys I I in j i what where where's the X go sorry I have it what's the I go along what's the J going along tell me something about Y and X tell me something about y and x and zal all orthogonal all perpendicular tell me something about I and J and K they're all mut we say mutually they're mutually orthogonal I and jk Are mutually orthogonal which means this if you ever dot product I with J what number you going to get have to if you ever dot product J with k or I with K they're mutually orthogonal you have to get zero if you dot product this you can think about it all right if you dot product I'll just do one example uh 010 which is J with 0 01 which is k um 0 0 0 0 see every time so these things we would call them mutually orthogonal and and it's it say this whole stuff kind of comes down to if you ever dotproduct I with j or or I with k or J with K and I'm not doing the reverse because we have commutativity um what are you going to get again if you ever dotproduct vectors that are perpendicular you have to get zero in fact that's how we prove that vectors are perpendicular how about this one for those you guys who are are thinking outside this take a little bit more thought a lot of this class I want you thinking I don't want to just give you this stuff that's that's worthless I want it to stick and the way that stuff sticks and I force you to do it yourself and force you to think about it I want you to think about for the next minute as I'm writing some stuff down what that's going to be think about what would have to happen to dot product this thing I and you can even write them out if you want to I DOI write out what the vectors are and try it see what it is write out J.J write out k. see what it is and then think about why why that is don't just sit here blankly for the next minute and a half okay I want you think it about it sh can tell you what we're about to take our break we're going to talk about this it's a it's a single number answer um and then during your break I want you to see if you can you can work on that we're when we come back we're going to go through very fast um because it's so basic you said you know a DOT product you said you know magnitude so we're going to crush the problem find the angle between them it's very very straightforward uh but this this is not so much unless you really think about it tell me something about I I they're parallel and J and J they're actually the same Vector they have to be parallel and K and K so if I dot product them all I'm going to get the same thing but what's the thing one one why why in the world when we dot product I with I or J with j k if you did it you you would find it out do 1.10 you're going to get one same thing with J's same thing with you're going to get one question is why well think back about it and then we're going to take our pause okay in order for two vectors to be parallel the angle between them has to be SLE correct tell me the magnitude of I right now What's the magnitude of I What's the magnitude of J What's the magnitude of K so if the angle between them is zero what's cosine of 0 what's cosine of pi negative 1 but still the multiple it's really inter one do that make sense it's kind of cool that's why we get this this one because cosine the largest coine we get is one when we have an angle of zero or an angle of Pi negative 1 but that means that we have this parallel idea have I explain things well enough for you guys to be able to do the cosine between angles I want you to practice that we'll come back in 10 all right welcome back so uh let's let's get this thing done listen it's not the hardest thing in the world to do but you you absolutely have to do it did you attempt to find the angle between these two vectors the uh this formula is pretty nice it lets you do everything basically at once so when I'm when I'm doing this when I'm find the angle between two vectors firstly understand that the angle between two vectors no matter how I look at that dotproduct is commutative magnitudes are just numbers and multiplication is commutative so it doesn't really matter the order in which I talk about this it's the same exact angle so when we find the cosine of the angle between them I start off like this uh you can also start off with theta equals cosine inverse that that's fine I don't really care I always start with this and then do the inverse later but here's how we do it the formula that we had said you're going to find the DOT product of these two vectors be careful on these vectors don't make sign errors obviously sign errors going to affect the angle all right so when we do this this says that and I'm going to write it out because I don't want to make a mistake the dotproduct of V and W since you guys are now pros at it0 cuz there's no eyes 0 * 1 plus -2 * 1 plus 3 * two heading on if you're with me on that one all this is is that or w.v it does not matter it's confusion still okay so far now on the denominator the denominator says okay we also need the magnitude of v and we need the magnitude of w and we're going to multiply those things listen you don't have to write out all of the steps that's fine just don't make the little eror that drive me nuts and you nuts but definitely make sure you have a square root on that thing this is not a magnitude squared okay unless you have the same Vector which you wouldn't because you just get a uh a one anyway because you well a magnitude anyway so here we go okay uh 0 2 is 0 2 2 is 4 3 2 is 9 so I know that that's going to be 0 + 4 + 9 do you see where that's coming from we've done magnitude enough where we should kind of be familiar with it we go okay hey magn to V 0 4 and 9 because we're squaring those components adding them and then putting them under square root heading out your p with that one times dot product or times it's only a DOT product if it's between two vectors if not you're just multiplying yeah it's the same little same Little Dot you should know what vectors are and scalers are at this point it's okay um the dot there means that we have just multiplication and then the other Vector 1 s is 1 1 squ is 1 2 s is 4 we got it under square root show hands if you're you're really okay with that one what's this again oh yeah it is scal but what's it coming from so if we do all of this that's zero that's -2 that's six how much do we get on the numerator denominator we have the square root of looks like 13 square root of yeah you can do a whole bunch of stuff with this okay you can multiply this and then rationalize it if you want to uh if you want to if you want to do I'm going to give you all the options that are acceptable okay so here's here's option number one firstly find Theta you don't leave it cosine Theta you find Theta how do you do that whole thing what do you do so if you do cosine inverse these two things if I've done it WR is 78 square otk of 78 that right there is an exact answer for the angle so what's the angle between them cosine inverse of 4 78 because it's not on a un Circle right but that that's the an exact answer you could do that you could rationalize and simplify uh if you simplify that one you get that that's nasty a lot of textbooks like to do that because they always like to rationalize me personally I do not care so if you give me this that's fine that's an exact answer give me this that that that's fine too that's the same answer uh can you approximate it can you approximate it in in degrees yes just make sure your calculator set to degrees and then if you do that just cosine inverse cosine inverse of whatever probably this one's easier to put in your calculator I think you get around 63.1 de so make sure you can do that on your own get 63.1 that's the idea I really need to know if this is enough for you guys to follow through and understand how to find the between any two vectors perfect let's try one more I want to make sure you're not doing a slip up that a lot of people do Daniel uh when they're going through this and I ask you to find parallel or perpendicular okay there's only one way to do this orthogonal perpendicular thing but there's several ways to do that I don't want you working too hard I'm going to show you the easiest way the go-to way are you with me on this one so number one thing if ever you're asked to find parallel vectors please only do this one way the only way vectors are parallel is if they are scaler multiples that's the only way that I want you to use there's other ways to do it you can use the dot product you can use the cross product which we don't know about yet uh we could try to make them on a plane and do don't do that scaler multiples so fast just do that one okay so when we're checking par whether these two things are parallel it should be pretty obvious it really should be pretty obvious so so for instance can you make one of these in one of these by by factoring out a number if you can they're parallel if you can't they're not that's what the scal of multiple idea is okay so I'll start with U with a and you go uh well wait a minute can I make that into that or could I start with b and and get that's probably even easier but could I make it into that yeah yeah look at it so look it uh it has pretty much the same thing going on there's a two there's a one there's a well oh that this has a three that doesn't let's Force this to divide out the three if I force this to factor out the three then this becomes 23 this becomes 1/3 and this becomes 1K did you catch what I'm talking about it's probably easier to start with this one and factor out the 1/3 but it's the same idea if you can factor out a number from one vector and get a number scalar times a different Vector that is how I want you to prove for all time that two vectors are parallel this right here proves it this says that a is a scalar time B and therefore a is parallel to B and that's the shorthand way that we write that I really want to make sure I've explained that well enough for you are you guys okay with that one show hands if you are for real do you see what I'm talking about might be easier to start here I just didn't if you start here factor out the 1/3 might be a little bit easier to do but it doesn't really matter find a number you can divide out get the other Vector times that number and it's parallel you with me now how about this one in actuality do we have to even talk about that if two vectors are parallel can they possibly be orthogonal it's kind of like different ideas right uh well let's prove that they're not how do you prove they're not easy easy any two vectors that are orthogonal when you dotproduct them Z zero your result has to be zero so therefore if I take the dotproduct of these two vectors and it's not zero they're not orthogonal are are you are you with me I wrote it down I said hey this the dot product is how you tell whether two vectors are orthogonal if it's zero they are if it's not they're not just go ahead and do that right now I want to see the dotproduct of a * b or sorry a. B that's how we say that A.B and then I want to see what that result is okay so go ahead in practice right now we already know they're not because we prove they are parallel did you get 4/3 plus 1/3 + 3 I don't even care what that is what's it not therefore a is not perpendicular to be that's that's what I want you to show that as if you understand how to check for parallel and how to check for perpendicular would you raise your hand right now in two words tell me how you check whether two vectors are parallel scalar multiples that's parallel in two words tell me how you check whether two vectors are orthogonal perfect that's exactly what I want you to know we do this one light speed okay really really fast um firstly look at these two vectors just tell me in like a second if they're parallel are they parallel with zero work are they parallel why not not CU to get from here you multiply by one and to get from here to here you don't multiply by one and you go oh no doesn't work they cannot possibly be parallel because they are not scalar multiples you guys get the idea it's that easy don't do any other work for paraller vectors that that's that's it that's it for real scal mile um are they perpendicular are they orthogonal well that that takes more work right you actually have to do something there's no easy check there uh unless you want to find the angle between them and then make sure that angle is 90° but that sucks because we made a shortcut for that do the shortcut find out whether or not these are are orthogonal what's the shortcut we just asked for it a little while ago how do you check whether two vectors are orthogonal do the dot product if it's zero zero they are if it's not zero they they're not for H were you all okay that they they're for sure not parallel because they're not scared of multiples it's pretty fast I mean it should jump out at you whether they are or not are you getting pretty good at the dot product it's going to get faster huh orthogonal yes no yeah absolutely these two vectors are orthogonal because their do product equals Zer and we just proved it the only way that happens is that the cosine of the angle between them is pi 2 that's the only time we get zero and that proves that we have orthogonal vectors here do you know that I have a dog I hate that dog I really he's a great dog but oh my gosh every time I don't we don't go on walks we go on drags because it's me dragging him everywhere cuz he likes to sniff and poop I don't know what's wrong with this dog but he poops like 13 times a day it sucks I I worn out like four shovels all right that's saying something so here's here's my problem all right we don't have a lot of Hills but there's a canal by our house which is a security risk and I don't really like that cuz people right behind our house on the canal so don't do that I have a shotgun ha what now just just kidding I would ever do that uh I do have a daughter though we're getting way off topic here so imagine that I get to drag I'm going to pull my it's a drag it's not a W uh I'm going to drag my dog up the hill for the canal okay I'm taller than my dog so even though I have to drag him up this hill so here's dog I'm pulling like this is all of my energy going towards dragging my dog up this hill no I'm also doing this to him so I'm also lifting him up this way dep on how I I I am pulling him and this is the depending on how so there's an angle between where I want to pull him and where I'm actually pulling him here's what we are going to do right now this is very Vector is there in this direction that that Vector creates the idea of the vector that V creates in the W direction is called Vector projection I love how that rhymes by the way the vector that V creates in the W direction is Vector projection uh it's just the directionality of that Vector it's nothing more than this idea it's nothing more than this here's a vector you don't need to write it just watch here's a vector true or false I can always break a vector into its horizontal and vertical components think of this as the horizontal think of that as the vertical it's the same thing it's just it's not horizontal it's a and different Vector does that make sense if the vector was not the x-axis in other words we're going to do that uh so that idea for the idea of of saying hey um what's that component of well what's what's the vector that that W make or sorry that V makes along W what's that Vector that that's a vector projection how it's denoted is that please please get this right I cannot iterate enough to you how many mistakes happen on this problem it's very easy it's not even hard it's just a little formula uh but a lot of people screw it up because they they have a misconception about what vectors projecting onto the other one the vector projection of V onto W is written this way projection so it's a vector projection the vector you are projecting is written here the vector you are projecting onto is written as a subscript that's what's going on that's the vector projection so what what this literally is I'm going to show you like just with a picture right now you ready if I drop a perpendicular I want the vector is is it's saying that's the vector I'm talking about it's not going to be the whole length of w it's not going to be all the way from from P to Q according to this Vector it's just going in that direction that's the projection of V onto W show feel okay with that it's a vector now every Vector has a magnitude the magnitude is what we call scalar projection why magnit SC we call scalar projection or component projection just the the part of it that's going in the W Direction so every vector's got a magnitude the magnitude of this one's called component projection basically it says hey uh how much of this Vector is moving along W what what's the length it would create there and it's written like this component vector v onto Vector W this Vector right there is that now we're going to work with it because it's got a lot of useful things but we're going to kind of we're going to build it a little bit what we're going to do right now is we're going to build this from the component uh the component aspect of it component projection and then we're going to get the the actual Vector projection from it and then we're going to do just a little bit of work with it not a whole lot you guys ready yeah you sure do you understand the idea of vector projection yes or no yeah let's wait for the proof and as we're going through the proof it's going to make more sense but then if they're still like ah I don't get it then then ask me some questions I can I can probably answer them uh don't sit through here going man I have no idea what this guy is talking about uh because that that's not that's not great okay if you're already there and like man this is the third day I have not I I don't know what you're talking about that's the problem all right you come and see me and and we'll get you some um but let's see if it's if it makes sense after this same exact picture we're still W to go my dog from P to Q okay here's the way I want you to think of component or scaler projection the first way how much of V is being applied in the W Direction that's a magnitude idea hey how much how much of my Force I'm pulling on that dog's collar is actually moving the dog up the hill does that make sense there's some wasted some's being pulled this way but how much of it is actually pulling him up the hill that is the idea of component projection so the amount the magnitude of my my V along the W Direction this guy is what we are talking about when we do this we just got to figure out some some ways to put it down mathematically now now there's something really nice about this you see an angle up there right an angle between two vectors and you also see a right triangle well we know we know some stuff about right triangles I hope in this class in fact we know that if I did a relationship between this and this what trigonometric function would relate this which is this length right here of the right triangle do you understand that the the length of that is how much of the magnitude of V is going along the W Direction that's that little length right there and we already have it it's a compon it's a scal it's a component it's it's literally a number what relates this to that coine relates adjacent to hypotenuse that's the adjacent what's the hypothenuse be careful before you answer what's the hypotenuse say say it louder magnitude of the magnitude of V because coine but side relationships don't work with vectors okay they work with numbers they work with scalers verify that's a scaler that's just the magnitude that's being pulled along W again this is not a scaler but if I took the magnitude it would give me the length of that just like a right triangle wants us to have did you know that we're almost done like we're really close can you solve for the component projection right now how do you solve for it h that's it how do you find you know what honestly it's a really trivial proof that I've just done I mean if you really think of about it this is any right triangle in the world it says Hey how do you find this length given this length than angle cosine cosine that's it it's just cosine because that's how right triangles work you go um hypotenuse time cosine Theta gives me the adjacent side that's literally all we're doing CH feel okay with that one that's it that's it now this also says something though it says um in order to find this you have to know the angle between the vectors what if you don't well so this is number one okay this is yeah I'll write it yeah I'll write it later too but this is going to be if you know the angle between these two vectors if you don't well but we know this this is pretty cool we know how to find the angle between the two vectors in fact we know that cosine Theta is um V do W over the magnitudes of those vectors so if we continue we got this let's just substitute it let's put this thing here it it's just algebra but the gist algebra can just kill you all right so I want to make sure that you're actually with me just a 20 second recap let me go through make sure you got it you understand the idea firstly that I'm pull that V is pulling something along W the amount of that is called the scal projection or component projection and it's literally just hey uh cosine Theta time this thing because that's how right triangles work and and that's our our definition of component projection so are you guys okay up to this far head not if you are yes what about this you guys understand that cosine Theta we just proved this spent like 30 minutes proving this are you guys okay with that one if I take this and put it here this V is this magnitude all this junk is all this junk do you see anything that simplifies that means that we have two formulas for component projection the first one I've given to you so the first way we find it that way that's the first way or why do we have two ways can you explain to me in your own words why we have two ways we have an say what now we have an angle that's right hey if you know the angle between your vectors and you want to do component projection which one of these formulas are are you using the left one or the right one for sure he's got an angle it's very easy man it's it's literally just cosine angle times magnitude done you guys with me on what I'm talking about what if you don't have an angle which one are you using this one has the angle built in there that's why we did it this way so this is like you know the angle you don't know the angle so you know Theta use that one you don't know Theta use that one I really want to know if I've explained this well enough because we got to continue here pretty quick we got we got to wrap this stuff up you guys okay with it you sure how about this one what what if the angle between those two vectors is more than Pi / 2 what's cosine do when uh when the angle of theta is more than pi/ 2 what's it do that's right remember all students so for you you guys all students take calculus right so all would be cosin's positive but then student second quadrant between pi 2 and Pi cos's negative does that make sense think about the picture if I am ping past past this past pi/ 2 what am I doing to that dog pulling it backwards I'm pulling him back down that hill so I'm pulling him negative the magnitude becomes negative does that make sense to you okay now here's what we know so we're going to continue we're going to do some uh we're going to change this one we're going to get to actual Vector projection and you'll at least have the formulas before you leave today we are extra I guess you know in the most basic sense uh vectors are really just a magnitude times a Direction here's what we know so this is just a little little proof about getting to Vector projection so if I'm looking for the actual Vector projection every Vector man I hope you can put it together okay every Vector equals a magnitude times a direction those words man they got to be up there what's I just did it we just did it what's the magnitude of vector projection there's definition like right up here I just eras it remember talking about how component projection is the magnitude of vector projection if if the vector projection is a magnitude times a direction this that is the magnitude now here's the question oh my gosh if you get it I'm going to go home so happy today otherwise I'm going to go home and cry myself to sleep like I do most nights give me the direction please just give me the direction just give me the direction what's the direction okay in this class you know some word association because I'm trying to brainwash you here Direction means unor unit Vector very good unit Vector of what unit Vector of V that's not where I'm pulling it unit Vector of yes that's the direction so Vector projection says hey find component projection because it's really easy and then go uh well what what what direction is it in whatever Vector you're pulling along man we're going along W right now the unit Vector of w don't do the vector W CU that gives it more magnitude than you actually have okay you ruin it unit Vector of w the direction of w what Vector you're pulling it along you're projecting onto I should say whatever this one is unit Vector of w here's what we're going to do we're simply going to take this piece I'm sorry uh lost we're going to take this piece we're going to put it right there and we're going to do one thing with it the only thing we're going to do with this is this here real quick check it out you see a magnitude times a magnitude right that gives you that magnitude squared idea which is really nice that's the best format to work with this end so that's the formula I'm going to give you do this first dotproduct with w and that will give you the vector that this creates in this Direction that's what that does we're going to work with some examples but could you find can you do it can you do it on your own right now so the examples are going to be a little bit extra okay could you find the dot product can you find a magnitude squared can you then dot product oh sorry scaler times a vector and then you're you're good that's the idea all right so now now's our time for practice what we're going to do is I'm going to show you how to how to actually do this it I do want it to make sense otherwise I would have just given you the formula and said hey do it uh but that that that sucks so I want you to really understand it so we're going to do this I'm going to show you how to what this means the biggest the biggest mixup ever like with with doing these problems I'm going to show you how it relates to work it's like physics aspect I'm going to show you kind of a tricky problem dealing with just a DOT product idea and an angle and then we're going to talk about um the whole unit circle idea but in 3D which is like maybe a unit sphere sort of idea that we're going to create so the first thing if you get what this is doing it says hey what Vector does V create along the W Direction well the length of that times the direction of w that's that's what it's doing it says hey you got this Vector it's going in this direction now the magnitude is just the component projection it's it's a triangle times the the vector that you're projecting upon the the the direction of that Vector it's a unit Vector just saying hey that's a direction of w yeah I know it's unit Vector that's how far I want to pull it in that direction and that's what's going on now how you do it it's not particularly hard but a lot of people get these two mixed up do you see how easy that would be to do it's very easy because when you're doing this you go oh yeah that's the same but then the unit Vector is completely different and this magnitude is completely different you have to get that right so the big letter we'll say this the big one the one you are projecting is the only one you use once the one and that should make sense the one you're projecting upon onto that's the one that has your direction to it that's the one that you should have the unit Vector with that's the one you're projecting upon which gives you your direction does that make sense that's how I remember it so when I look at this the the first thing I we're going to draw two pictures and we're going to do it it's going to be pretty quick after that I'm projecting onto a a should be the one regular unit Vector which means a should be this and a should be this B is the only thing I have right there does that makes sense if I'm projecting on to B vice versa B carries my direction B is the one I get my unit Vector from B is the one that goes here and here and that's the way you think about it if you just have the formulas it's very easy to get confused that's why I teach you on you need to understand it uh because if you understand that the one you're projecting onto carries your direction the one you're projecting onto carries your unit Vector the one you're projecting onto is what goes here here that's important feel okay with that one that's a big deal it's a really big deal now let's practice so the first thing how this looks if I have a vector a and a vector B what this projection is doing is the original one it's the original one we talked about it's taking B and dropping it down to a and seeing what happened that's what's going on it's dropping B to a you guys understand the the interpretation of this I'm going to do the other one right now just so we get it clear in our head I'm going to draw try to draw the same picture this idea is different notice the difference there not the same this one says I want to project a in the B Direction onto B do you get the idea this one dropped B onto a this one climbs a onto B well in order to do that here's what happens he goes oh okay let's um so make it perpendicular wait there's nothing there yeah it's a way bigger vector it's going to be long even when I project it it's going to be longer than this so what's really happening is saying well just give it somewhere to project upon and that purple Vector that's the projection of of a on to B do you get the idea it's it's just guys it's it's honestly all about how much it is component times the direction your headed unit Vector this is headed along the a direction this is headed along the B Direction they're going to be different vectors they have to be different vectors you guys see the difference between them okay now it's just a matter of using the formula it really is just practice so I'm going to do the first one with you I want you to do the second one on your own I'll give you a couple minutes to do that but but let's practice um the the main idea i i t T you the main idea uh the main idea is goodness gracious know what you're projecting onto because it's really easy to mess the formula we're projecting onto a a carries the direction a carries the unit Vector that means a is this piece right here a should be there I would write that out I I don't want to do this in my head I want to write the formula out with the correct letters so that should be b. a you can't screw that up I mean you just do product the vectors but this you can what what should this be should that be a should that be B what should it be it should be what you're projecting onto here we were projecting on to w we had W here we're projected onto a you need a figure out that first the reason why I gave you this this is the best formula for it okay this one reason why I gave you that is because this is going to be a scalar and then you just multiply it by the vector a and it saves you just a little bit of work not a lot but just a little bit that's what we're going to do show fans if feel okay with how that formula works why we have the A's where we have the A's yes let's try the rest of it's going to you it's not not too bad we're going to do a do product so we have 2+ -2 + 3 check my work too I'm I'm I'm great at making mistakes on this stuff sometimes they do it on purpose just to see if you're listening did you get the same thing I got yes this is all going to be a scaler it could be a number now this one man use those that's why I love this formula because it's that magnitude squared is really nice it says you're going to look at a and remember the magnitude squared it's the magnitude without the square root it's really easy it's you square all the numbers and you add them just don't do the square root that's why I give you this formula okay a so 1 4 9 done cuz you're squaring them it's the magnitude squared you just don't have the square root does that make sense to you yes what goes here it is a vector scaler Vector that's okay but this is a vector oh my gosh uh looks like three what now 3 14 let's do a little interpretation I want you to leave this hanging here here's what this says it says that the part of vector B that is going in the a direction is this Vector right here that's the vector of it does that make sense if you wanted to find the how much the length of it you'd find the magnitude of that Vector right now that would be the component projection or you start over because the formula is really easy and and you do the component projection but that's the interpretation show pans interested in the interpretation of it perfect can you do it 10 points well Five Points you got to do this one too but 10 points on the test I mean that that that's that's really it um sometimes of course you're going to have to distribute this if so if your answer is not exactly the same probably distributed that but for me I I really don't care what I do care about is that you can do that one to get right go for it for for oh man you do it did you get it did you get this one yeah first thing I'm doing for real like first thing I always do because it's so easy to make mistakes on it I read up the formula with the correct letters and the correct spots do you see how that would be pretty useful to do is the the formula you're going to have that formula well I erased it you're going to have the one that used to be here just think back you're going to have that formula on your note card that you get on your test you need to know how to plug that in though I'm not going to give you v's and W's or maybe I do but I give them to you in different ways because I'm a jerk H you have to understand it okay so for us we go okay well well what this what this says is if you understand what direction you're headed in and you get that that gives you a unit Vector it tells you what letters go where that's the idea of understanding this not just because it's your only other choice CU you're switching them but hopefully you understand we're in the B Direction you guys okay with the idea please be smart about it don't reinvent the wheel all right a and A.B b. a is the same thing so this is going to still give you that right there yeah you're going to have to do a little bit of work hopefully you do the correct work yeah I think so because we have square square square no square root because we're squaring the magnitude man you've learned a lot today tell me one thing you're never leaving on a test for me please don't do that simplify your fractions so we're going to we're going to change this to our 1 half in this case it might be nice to distribute that I don't I don't care all right I really don't I don't care um most textbooks will uh the only time that they really don't is when you have the whole unit Vector idea and then you grow it into another Vector remember doing that uh you multiply the magnitude times the unit Vector so with projections a lot of times they're going to and that's most likely what you're going to see have I explained this well enough for you guys to Ace this on a test good I hope so I hope so you want to talk about some work not that this wasn't work already but work in the sense of of of what work means for for us in um in physics that sort of idea what work is how I like to think about work it's a component of a force which is a vector a component of a vector directed along a path in a certain Direction that's called the directed path another Vector um have we just figured out how one component of a vector is directed along a different it's it's projection it's a form of projection uh the only thing that is that work is defined kind of specifically it says that it's it's multiply which is what we've been doing says that we have just watch you're going to see everything that we've just done okay if if you're still awake out there um it says that we have we got some Force right which is a vector that's the magude magnitude of the force come on just put it put it together that's the angle between the force vector and the path Vector Force Vector path me dog Hill okay that's the angle between the force vector and the path Vector this magnitude times this is if you think back to I I don't have it on the for but it's on the video look back okay look back at your notes look back in the video you're going to notice this this is component projection that's exactly what that is does that make sense the only difference between what we've been doing and what we're doing now is we're saying we're saying this this right here is the component of the force along whatever Vector I want this this thing to move okay my my D my displacement Vector okay come on dyslexia I don't along D the only difference between what we've done and what we're doing now is that work is defined like this work is not a vector work is a scaler it's how much you got done okay so what happens it says hey um this is the component this is how much the force is moving so how much we're applying in this direction times how far you're moving it which is the complete displacement which okay come on now if I have a vector that says I'm going here to here and I want to figure out how far that is tell me in terms of vectors what that means okay distance tell me in terms of a vector what that means it's a magnitude yeah work is a component of a force boom we got it component projection Force along the directed path times how much you actually move did how much you did that's what work is defined to be does that make sense on what work is right now do you see how it relates to what we've been doing it's kind of cool can I just blow your mind to some other stuff here can we check this out scaler yes which means it's all communative which means that work can be reassociated oh but wait when I have a magnitude of a vector times the magnitude of another vector and the angle between those two vectors this thing look back at your notes you're going have to look back at your notes okay I'm bored but there was a definition here I said hey remember that uh there's an alternate way to do this and it's really useful I'm we're going to talk about it when I do this little proof here it is force. displacement equals work do you see it do you see it you remember that little definition if you think back to where's you don't have it anymore but if you think back to here we had uh we have that do you remember that we said and right before we did that we had this stuff on this side and that that was this that's what I'm telling you so because we have this definition of work where where work is um the component of force of directly along path times the amount that you're it that's what work is they go okay that's fantastic wait a minute that's just a DOT that's just a DOT product so what is a DOT product work work and sometimes it's some work that's pretty cool that's pretty cool here's the deal just just like before okay just like before if you know the angle between the force and the path you're moving stuff on which formula are you using for work this one because it has an angle right built into it okay if you don't know the angle which one are you doing that one so you could do either way I not if you're okay with the idea tell you what we're going to burn through two examples pretty quick um we're going to do one where we don't know the angle one where we do know angle we're going to get to my dog example uh and then we'll we'll talk about that the whole unit unit Vector in 3D idea so okay so first example here's what's going on I got a force Vector so I'm pulling along this Vector I'm pulling along that vector and what happens is I'm pulling along that Vector but I have this little Runner okay it's like a track and this track goes between two points it'd be a pretty lame question if this Vector was between those two points it's not and it's not going to be parallel either that's the whole point of doing this okay it's saying I got this little runner between these two points wherever they are in space and I've got a force Vector I don't even know what that doesn't have to be right on top of that that's just that's polling it's going to eventually move from here to here what we want to figure out is how much work is done when we do that because we're we might be applying a whole lot of force in a wrong direction it's still going to move it but we're we're taking a we're doing we're doing a certain amount of work here so we're going to figure that out do you guys understand the idea on this tell me what you have tell me what you don't have we have points you have points do you have a force Vector do you have a displacement Vector yes explicitly written out can you find it oh good okay so the first thing here is okay which one am I using do I know the angle between them no I could find it but that's going to suck so why because we have a better formula anyway we need two vectors to find out the the work done by moving a point from one spot to another on them we got to have that so the first thing we have is this Force vector and that's great the next thing we have to do is figure out what our displacement Vector is if you're given two points man your displacement Vector is the position Vector of those two points find that now okay my handwriting is just going downhill sorry did you figure it out the displacement Vector what is it tell me an egregious error I just made that none of you are going to make because I'm telling you right now why isn't it a vector because all I'm giving you is the terminal points okay and I don't want that so when you're talking vectors you need a vector notice if I try to do a DOT product right now with a point that doesn't make sense I got to have vector make it make sense egregious all right so now I've got a vector Force I got a vector a displacement I can do this one of two ways easy way or hard way I can find the angle between these using all that junk that we just did and then find the magnitudes and use this formula right there or because I have a force Vector because I have a displacement Vector tell me how you can find work do it find the work for for that's pretty good best drawing I've done all day only drawing I've done all day but you know whatever can you picture the dog yeah it's a it's a wiener dog I don't even have a wiener dog there's one that comes around gets in my garage and walks on my maple top plywood that I'm making cornhole boards out of pisses me off best I got sorry I'm going to give it away I don't even know who is say hey want you guys want a wind doog cool I guess stupid Wier dog so yeah I'm walking someone else's Wier dog now yeah thanks thanks for that show my Gage a dog did you figure out the work on this if you know how to do a DOT product yeah that's not not a problem no problem so we're going to Dot this 6 9 -3 12 12 units of work whatever those units mean did you get the 12 yeah awesome any comments questions anything about about that one let's do the dog example okay so I'm walking up this canal this canal is incline of 20° this canal is a long I've never heard of a 30ft canal but hey let's just pretend it happens so I'm moving this dog 30 ft and I pull up an angle of 30 I'm estimating here so we're estimating work I'm pulling up about 30° about 15 PBS it's he's actually like a 70 lb dog so I'm hoping I don't have to drag he's head how much work is that number one thing I want you to think about is what formula makes sense here does it make sense to do a DOT product on this example no a DOT product means you got to you you you're not given angles all right and first do you even know what those vectors are how are you going to do a DOT product if you don't even know what the vectors are I can't do that you're right you can't that's why we get this other one look at this form look at this formula this is really a beautiful thing do I care you're not even looking you should be looking do I care what the actual vectors are no all I care is the magnitudes do I know the magnitude of the force how much 15 lbs good do I know the magnitude of the displacement do I know the angle between them what is it what about the 20 matter it's does the 20 matter that always screwed me up in physics like what about the 20 it's got to be there for something it's not it's not there for anything it's just telling you okay it doesn't matter uh that doesn't matter when we're moving it along this Vector it does not matter uh what we're talking about here is okay I have 15 pounds of work I'm not talking about gravity right now we're saying I'm moving this dog it's that's enough to move him so that's what we're talking about so first thing okay work did I make it no say yes they can't see it yes yeah so work uh no because I don't even have vectors all right work goes yeah let's let's talk about the actual magnitude of the force the mag the actual displacement and then the cosine of the angle between them we just talked about we're going to burn through it pretty fast the force is 15 lb that's the magnitude the displacement is 30 ft the the angle between them 30° do you know what cosine of 30° is yeah root3 over2 that's the exact so we go okay I think that's 450 if I do a math correct yeah think so work is about equal to so take the 450 multiply it by cosine 30 or multiply it by square 3 divide by two how much do we get roughly 3897 yeah that's about what I got that's a office sign units kind of matter in real life units units matter here so 3897 foot pounds you the foot pounds moving a pound of we moved 39.7 foot pounds you ever talked about torque if you ever worked on any sort of vehicle whatsoever and you put on like a lug nut you got to torque them to a foot pound we're going to talk about torque in the next section about how we actually get that stuff has this made sense so far yes perfect any comments questions whatsoever okay we're going to talk about something that uh not super used all the time but it comes up occasionally uh and because I talked about the unit the unit circle idea in relation to a unit Vector I want to bring this up so last little thing kind of big thing but last thing I want you get a feel for it a feel for how the angles work on a vector in 3D you know how the angles work on 2D because it's really easy you go uh Vector angle duh but when we talk about 3D we have these things that are going everywhere right from the origin I want to I want the only reason why we're doing this cuz I want just start picturing the angles in relationship to the planes that's what I want and it's good for you so we're going to talk about something called Direction cosiness what direction cosiness do is is tell you how to relate where Vector going to each of the axes X Y and Z that's what they do how they do it I'm going to show you can you please draw yourself a 3D coordinate system do you know how to do that yet first draw straight lines I'm not drunk I just don't know how to draw good straight lines so draw yourself a nice good straight line I ever tell you that story about when I found out this teacher doesn't matter was actually drunk in class it's crazy and it wasn't me which is shocking all is that a three coordinate system did he fall over no I got to to it right now got a raise promoted anyway which one of these is the x-axis yeah it's that one that one that's the ex it's going into the board right now it's a little awkward because we're trying to represent 3D on 2D and you can do it but it's it's you have to visualize this one what is it and you should be familiar with what the positive and the negative x y and z are here's what we're going to do we're going to do is take a random position Vector like that one that I want you to interpret as not being on the plane it's sitting out here in octant one can you picture that right now so it's it's not it's coming out like this that's what's happening here's what we're going to describe right now we're going to describe the angles that this Vector makes with with each of the three axes so here's how to picture it appropriately so that your head doesn't explode okay we're going to first talk about the angle this would make with the xais trust me you gota you got to listen right now otherwise is not going to sink in Imagine This Vector making a cone around the x-axis can you picture that right now making a cone around the x-axis it is going to intersect the XZ plane and the XY plane and if you imagine that it's going to have an equal angle no matter no matter that angle is going to be the same on those planes you guys get what I'm talking about so imagine a cone going around the x-axis the angle we're talking about is the angle that that projection makes from X to the picture on the XY plane or from X to the picture on the XD plane of that cone that's the angle if I imagined it being a cone revolving around the Y AIS that thing's going to intersect the YZ plane it's going to intersect the XY plane the angle that's a second angle is the representation so if I just make the cone that that angle that's what I'm talking about that's going to be the angle with the y- axis are you guys getting the idea Imagine The Cone going around the Z AIS is going to intersect the XZ plane it's going to intersect the YZ plane that angle that it makes with the picture of the cone is the angle I'm talking about with the z-axis you guys get the picture it's hard to visualize but you visualize the cone and it's a lot easier okay remember we talked about dogs right you ever seen a dog with a little cone on his head that's what I'm talking about it's going to intersect here it's going to intersect here that angle is what we're finding and there's three of them there's an angle that does this this one's the where is one from the X to the vector we call that Alpha from the Y to the vector is beta and from the Z to the vector that way that way the it's always from the positive axes okay it's not from this way it's from the positive a that that gets I'm getting used to okay that angle measurement from the Z it's dropping it's not climbing because that' be from the negative it's from the positive or what do we call that one I think we call it gamma I'm horrible at gamas by the way are you horrible at gamas that's the best I got actually it's not bad for a gamma me cuz I'm not Greek did you know that now you can tell but whatever alpha beta and gamma are called the direction angles of our vector you ready for it you ready ready for the first question everybody in class right now tell me the direction Vector of the of the x-axis tell me what it is the direction Vector the unit Vector that represents the xais I thank you I oh I see now mhm uh I get it let's try it again maybe you can get this one everybody in class right now tell me the direction Vector of the y axis oh Geniuses that's right all of you uh do the direction Vector of the z- axis what's that K perfect if I want want to find out the angle that a vector makes with a partic with another Vector all I have to do for that cosine it's called the direction coine uh all I have to do for that that angle is dot product the vector and the the vector that I'm given and the vector that I want to find the angle between if I want to find the angle between the vector and the x axis you just told me right now that I kind of missed still not drunk don't worry about it I just missed my line okay uh you just told me right now that I is that direction does that make sense are you sure you're with me yes all right you guys okay with it do you see where it's coming from that's where I really want to know do you see where it's coming from this says I want to find the angle between V and I what's I oh the x axis that says that this is now going to give us I'm sorry for erasing that but that that's going to give us Alpha the angle between V and I is what Alpha is defined to be did you guys get it now let's make it even easier and it's going to make a lot more sense make a lot of sense we do this let's make it even easier now let's actually dot product V and I I'm going to do it over here but I'm going to erase it okay so let let's do that here's V yes no you sure R Vector right you need to know this what's I come on quickly someone said it earlier but everyone say it what's I let's dot product them you all know how to do it you multiply and you add um what am I going to get out of that dot product just V1 so this is simply V1 not a vector just a number because it's do product SC just V1 so understand that concept now how about this one how about the magnitude of I we need to know that one what's that one magnitude of V I don't know that's the only thing we really have to work on but here's here's what we just learned about the cosine Alpha the the angle that the vector mates the xais cosine Alpha equal V1 over the magnitude of V please stop what you're doing right now just just listen for like 5 Seconds okay this should make absolute sense just by the logic of the cone and what a right triangle does think about it if I had this cone and it moves it right here so this vector's coming up like this um the magnitude of the vector that's the length here drop a perpendicular cosine of the angle takes hypotenuse over adjacent oh sorry um I had adjacent over hypotenuse takes adjacent over hypotenuse if you solve it for that that Co it's that's what this is that's the adjacent look look at the look at the adjacent the adjacent is the X component you guys see it that's the X component of the vector divided by the magnitude that's literally what a right triangle does this is adjacent over hypotenuse done did you guys see it lot of proof for it but it's just adjacent and so what that means for us for cosine beta I'm not going to do the whole proof again because it's really easy if I wanted to find the angle between the y- axis check it out this beta I just pretend it's on pretend it's on the plane it would be adjacent that would be the Y component over hypotenuse the length of the vector magnitude and likewise for gamma yellow ke on the idea behind this for real we're going to do maybe just like one or two examples we'll call it good uh there's one other thing that I need to show you and then that's that's it first one's that first one's that I'm going to prove it right here with just some some talking through you know what it's too easy to I'll just prove right here I passed that to you guys' rule right second time yes okay good hey think about it if cosine Alpha is this and cosine beta is this and cosine gamma is this and you square them all look what happens look what happens I get magnitude squared magnitude squared magnitude squared as a common denominator I get V1 V2 V3 all squared respectively which says okay this is uh that but we we learned something earlier come on everybody everybody you got this what is that that's the magnitude Square that's one that proves it right there so that's a really neat relationship that we can use on these things um if you ever have to find the unit Vector that's a quick way to do it here's how you find the unit Vector it's just like finding the unit Vector from the unit circle idea it's just a a little extension it doesn't have any signs because we didn't use sign up here to talk about the the relationship between the the axis and the vector that's the equivalent that's the way you can find if you know the direction angles that's the way you can find the unit Vector does makes sense for you do you guys want to do an example or do you feel okay with it you want to do one example yeah yeah okay can you use this to figure out what alpha beta and gamma is going to be yes first thing I want you to do just find the magnitude because you're going to use it in all three of those cases so number one thing I'm doing is magnitude of of vector B not magnitude squared you got to have magnitude so I know I'm going to be working a square of 35 a whole lot okay we're going to burn okay we're going to go really really fast cosine Alpha you tell me the component that I need for cosine Alpha is it X is it Y is it Z it's really convenient right because Alpha has to do with the x axis it's the X component so tell me specifically what I have over could you find Alpha from this yes how would you find Alpha from that notice something that's going to be more than 90° more than why well cosine negative that that happened in the second we can't go third or fourth all right because that's that'd be that'd be too far around we'd get there shorter the other way also notice notice this this Vector it says hey on the x- axis you're going You're Going negative you're going this way you're going more than 90° that should make sense when you get that that negative number it's more than 90 because you've gone past the Y AIS you're now in Negative X land does that make sense cosine beta you you can do it it's 3 35 and then cosine gamma 5 35 you can do the cosine inverse and get nice approximations of those things have I made this stuff make sense for you yeah I'm just a little unclear about the negative cosine Alpha how it correlates to that you said it goes over 0 imagine if you on your vector you went your first step is going 1 correct 1 so1 is this way yes in order to make that work with my x axis that angle that's going past the Y into negative xan that's more than 90° that's what I'm talking about here the other two are going to be less than 90 cuz in positive y land I'm this way and positive Z land I'm this way I'm not past it does that make sense yes pretty cool uh anyway that's that's about it