so back at we're going to we're going to go not really switch gears we're going to take this and we're going to we're going to change to D degree and we see how that relates and you're going to like it because a lot of what you do in to D has this little extension in 3d and it's it's not really anything different it's just you have this to this third dimension now if you didn't know this mmm a lot of our world sort of like takes place in 3d did you know that that's crazy I know right well which means that one plane really doesn't cut it it would cut it if we were always just on a flat road driving a car that would probably work for cars but we have like planes right and we move we have particles that look about we need to be able to describe them in 3-space 3d are three all this stuff so we need a new coordinate system and here's how we do it please get this right it's not hard okay but it's it's a little a little different what we're going to do is instead of just having our XY plane we're still going to have that but we're going to do is instead of having our XY plane up here we're going to we're going to drop it so our our plane falls down horizontal and we get this this different axis so XY seems logical it's a mouth before and then we get the z axis going perpendicular to both the X and the y so x and y perpendicular disease also the mutually perpendicular that's a 3d coordinate system so when you're drawing it you still draw this but please please it's not X&Y anymore this is not x and y the X is a little weird the X is the one that comes out toward you so the X is the one who does this this right here is the x axis this right here is the y axis and it has to be the Y its it's laying flat and if you look down it creates the XY plane for you like it like it normally does do you guys see what I'm talking about it's the Z that shoots up please please always write the 3 to the 3 3 plane coordinate system like this please do that now we're going to identify a whole lot about this before we can move any further the first one is where are positive numbers and where are negative numbers so here's where they are the positive numbers for X are coming this way so this is the positive x and this would be the negative x so here's like 1 2 3 4 negative 1 negative 2 negative 3 and negative 4 the one where do you suppose the the positive Y is to provided your left yeah we don't want to switch everything around the positives is to the right um on the y axis so 1 2 3 4 this is positive Y and negative Y is over it moved and the Z the same concept positive Z is going to climb negatives easily involved we can know if you okay with that one so far it's a little funky right but I mean it's not 2d anymore so I've got a way to represent three dimensions this is the best we got so this by the way is called the right hand coordinate system here's here's why it's called the right hand if I put my if I put my hand out where my fingers are going along the positive x and if I curl them towards the positive Y my thumb is pointed up towards the positive C do you see what I'm talking about but I can't if I put this harder my shoulder but like then I go positive X to negative Y well that's that's pointed negative C does that make sense to you that's what's called right-handed so positive x to positive Y gives you positive Z X to negative Y gives me negative Z that's what ends that's the notes your right hand left-handed would be when I went positive x positive Y we give me it be by the way why don't we shift it another give me this we're talking about right-handed remortgage system this is generally what we reveal so do you feel comfortable about the positive negative axes yeah now could you could you identify the XY plane could you could just show me like your hands like how it goes what is it is it going is it on the board is this the XY plane now how's the XY plane yeah that's right it's this flatten that's kind of cutting into the board how about the the Y Z plane what's the Y Z plane is the Y Z plane cutting into the board or is the Y Z plane on the board or Y Z plane is the bullet the XZ plane is the one that's going things before this plane I'd knock you okay with the planes all right fantastic now points points can be X and winding so hey I'll plot the point two three you're going to go wait a minute two three where's two two with the X and 3 would be Y where is that going to be though because I need that third thing now r2 so like 2d you just need two points are three you need three three values three chords so we know how XY and Z it's always alphabetic order I got to be honest with you it's so they're ordered triples not not ordered pairs they're not the easiest thing in the world to to plot here they really aren't because it's weird to go outside of a plane you know I'm talking about or you don't yet but you will you will it's weird to go outside of a plane so we're going to try just a couple so you get the idea on it I'm also going to talk to you about something called with my quadrants how many how many sections are there if I used my planes how many sections what I have there was four quadrants as I went why two three and four and now each of those quadrants has a top and bottom four quadrants HMS top and bottom eight eight octants one all two three four so it models the quadrant system one two three four and we drop down by six seven eight so above quadrant one they've opted one and below we have opted in five does that make sense to you we'd go around like that so let's try to plot just a couple of them the first one I want to plot an easy one let's do two zero and negative three okay everybody quickly really quickly what is the x-coordinate what is the y-coordinate is what's a Z coordinate here's how you do it's always that order so from the origin we have positive two that's here that's it's here y is zero so we don't go anywhere and then we drop down how okay now it's a little weird here's the best way that you can plot these things it's going to take some parallel lines here's how to you're so dry because you aren't artistic I'm not either but if you got like a ruler and can make parallel lines this is the best way to check this out so here's the the to look just a little temporary mark here's the zero on up move it here's the negative three on the z-axis you follow I'm talking about from this point you're going to do two things you're going to draw parallel with his z-axis this way and parallel with the x axis this way and where they meet that is where your point is watch so I'm going to draw a parallel that looks pretty close to parallel for me on the X x axis does that make sense to you it goes through the Z coordinate and then this I'm going to draw parallel to the z axis the intersection of those that right there is the point 2 0 negative 3 it's floating in space is on the XZ plane because the y coordinate is settled so it's right there on that plane and now if you're given column do you see a problem with a little weird graph right it's going to take practice there's no other way that I can tell you but but it's going to take practice it's good to take we do one more with you and after that it really is just all about practice okay let's do help a 2 3 5 2 3 4 quickly export a place like morning very Z chord what often will this be in I go out right they go out on the X because positive then I go here and then I go up that's it that's that's lo me that's floating up here that's like right here okay so it's a work we're an often number that's where we're at let's try to graph it so when you're has when you have these do the X on the Y first so what we're going to do is we're going to use our parallel lines for I'm trying to show this to you that way you can actually visualize these points here's how it works draw your parallel axes again so from my two and from my three I'm going to draw parallel along the Y this is the hard one it's the x axis drunk parallel so it's like this both parallel along the X do you see what I'm what I'm doing parallel along X and parallel along the Y I know if you see what we'll be talking about hello yes no kick true/false the point is right there true or false no false yeah this is the what we call it projection if you were to look down upon the world and the XY plane is flat the point is raised up here but it would look like it's right over that spot on the XY plane does that make sense to you like a bird's eye view that's where it would show up that would be the shadow or the image or the projection that's what's happening now where is this point really though where where is this point is it up or is it down how much is it up so from right here you can do this two ways you can just take this amount and go straight up parallel that's typically how probably do it but you lose a little bit of measurement that way so go okay my hands pretty bags are there there are out there that's where that is but there is a slightly different way to do and I'll show you that way as well another way that you do this is you just because if you if you don't know if you don't know where it's at you extend this up as far as you can and you do this this is a weirder way that's more accurate draw yourself that segment that intersects the origin with the projection of your point are you following what I'm talking about and make it dotted here go up to the appropriate height now what's the appropriate height what's our see here okay hi over here shift this up in other words make it bearable and where that intersects that's where your point is in space big gives you a decent visual representation about where you can you guys see the 3d representation I'm trying to draw here so it's it's not exactly down here it's shifted up but that gives you a good frame like oh this is weird is this on a plane right but I got a show that it's not so I show my parallel lines and that's a good way to do it that's a man I take a while right it's going to take just practice for you to for you to do this and not if you're okay with at least one of those ways of graphing lines perfect what about this what if I said I want you to grab x equals two could you do it what is x equals two lines would be a 2d what's a lying in 3d what's a line that goes forever in a plane if we have x equals two lines now become planes circles become spheres okay functions become surfaces so we're extending that the dimension so when i say x equals two that x equals two says x equals 2 or everything that's a plane so the x equals two would cut through the x axis at two and go all the way this way how about y y equals five if I did y equals five what that would do is we come over here five we cut into the board at five it would be like this looking like this because that would intersect that would do just just like that x and y XR x and z could be anything that would cut right through y equals 5 is that make sense can you visualize Z equals negative 2 what plane would Z equals negative 2 be parallel to perfectly XY plane if you like take an XY planet drop in it two spots that's what Z equals 8 are you still with me am i boring you to tears I hope I'm not it's not super difficult but it's super-new which can be difficult the last thing I want to talk about before we take just a short break when we get an equation now and when our equation is an X's and Y's 2d we got forget lines and stuff right my functions but when we get X's and Y's of Z's now we now have the ability to shift through space and the x direction and the y direction and the Z we don't get lines necessarily we can and we don't get plays and so that we can look at these things hold surfaces we get weird stuff and it's pretty cool and that's really the focus of this class is how we deal with the calculus of services and things that are on services and traveling through surfaces stuff like that that's the majority of and then we talk about vector fields and so that's later on but let's go ahead and take a break and then we'll come back and fill out this all right time to change our 2d stuff and the 3d stuff okay we're going to start here you're going to find out that all the formulas that we use for 3d coordinates are based on 2d corners they're very very very similar this had the third component so so for instance our distance formula if you want to find the distance between two points and you had a 2d system this is the distance formula for 2d x2 minus x1 squared by 2 moments quits he'll become magnitude if we want the third it's literally just adding on the z2 minus z1 squared that's all there is to it which means this now think about this kind of thing in the future when we do vectors notice how the magnitude the distance formula is basically the same for 2d and 3d then the magnitude is basically the same for 2d and 3d and that's living in us does that make sense to you so we're going to get that for me now it's just a little short example do you know what an isosceles triangle is you know what that means one has two two sides that are exactly the same so if I wanted two sides have the same length I had three points well I could find the distance between every set of two points and check those distances so the idea behind doing problems like this is try to try to think back to the geometric aspect of it and and do the this is formula do the midpoint formula do what it takes to compare these these sides or something so if we found the distance from A to B the distance from B to C and the distance from A to C that would represent four three two sides of our triangle can you guys do that because you find the distance between a B and a and C and being sneaky to do that let's try it let's try it real quick I'll do the first one for you did she show you how how fast this really goes but the idea is you should be able to use the formulas then if I ask you for something like oh hey is it a right triangle well if you can find the distances and they fit the Pythagorean theorem then you'd have a right triangle is that does it make sense were you listening I'll say it again later then okay if I'll be working so the way it works just take the x2 minus x1 squared x2 y2 minus y1 squared z2 minus z1 squared so from A to B we have 1 squared plus 0 squared plus 5 squared this is going to give you the square root of 26 can you verify that over square root of 26 do you see where the numbers came from I don't want to blow past you I don't want to go too fast for you are you are you seriously with me okay how about the BC what's the first number I would square one do we have to do negative one when we're square and stuff because if I want BC it really be three minus four does it matter I can go see B it does it does it definitely matters so if I want to do here to here that's fine too that's probably a little bit easier so we have one squared plus three squared plus what's the next one sixteen nine one true or false it's at least isosceles for absolute why because I have two sides of the same so that fits isosceles the third side I think we can square root of ten but we can do it ac is zero squared plus three squared plus negative one squared when you use t1 squared some of the things that a lot of textbooks asks for are so low is it isosceles can you verify it is isosceles just oh is it a right triangle well then you do things like well do two of these sides squared equal the hypotenuse squared it doesn't here so no it's not another right triangle but you talk about my favorite theorem if I said are they collinear do two of them add up to a third one well that would be collinear if two lengths add up to the third one you don't have triangle anymore you have an on triangle you just have three segments lying on top of each other that that's what happens here does that make sense to you that's the idea but it uses some of this stuff midpoint formula is just an extrapolation as well if I want to find the midpoint this is midpoint for two points we're sorry for a 2d system it's just averaging accordance and that's exactly the same thing we do for for three now we talked about it briefly but we're not going to have a whole lot of circles anymore if we take the idea of circle for 2d what shape do we get 4 4 3 News yeah so circle is now changed to how much spheres what's really awesome up sphere just everything you learned from circles pretty much works for spheres just like distance just like midpoint we just have this little addition this little concept added on to it this right here please watch carefully this right here would be the equation for a circle this is it all we have to do is add to it the third dimension what I use - though the only thing you really getting down to like center of a circle it's always the opposite of whatever that sign is so if this is a - the center is a positive number that's a plus the center has a negative number in it does that make sense to us as simple as I can make it so our Center for sphere is very much like the center of a circle you have age okay and we're going to have L the radius actually still works also the radius is just R which is nice want to click head on if you're ok with with the so far be sure that's right an example here real fast all right let's go ahead and I'm going to give you pretty nasty equation we're going to see what this thing is well that's a lot that's a lot of stuff number one anybody want to do that no no it no I'll do that's a lot it's not a thing that's a thing here's how to cope with with a lot of this stuff number one you're going to get pretty used to looking at an equation and giving me a surface that's associated with it trust me you're going to get very good at that because a lot of things that we do in here deal with these services a lot of quadric surfaces that we have so we'll talk about about this thing but here's some things that you can try to see if this thing is a circle or us are a sphere or cylinder here's one of the concepts number one thing if you have square square square same number no other squares probably a sphere all right it's probably going to be some sort of sphere or Lisa quadric surface in that case when you have squares for square same exact number let's go ahead try to group or variables let's try to get no coefficients in front of those squares because we're going to try to complete the square that's that's the idea here so let's try to get a look at let's move our constant we're going to also group our variables so what I talked about that I mean let's put our X's together let's put our Y's together let's put our C's together and let's give our constant off to the other side now if you remember completing the square now this this is kind of how I do review with you I do problems that deal with some stuff that you should know it's going to give you a little chance of review all right so if you mail how to complete the square that's awesome if you don't know how to complete the square the idea is you can't have numbers here you can't have numbers here this is why we look for the coefficients of our square terms to be the same because we're going to divide those out so we're going to divide everything by two I give you next problem divide everything by two and I also mean this sign so go ahead and do that now you'll know if you made that far gimmick we cannot get made that point cool do you remember how to complete the square I'm going to do one of them and your results for the other ones okay here's how you complete the square what happens it take this number you take half of it and you square it so when I take out this number it's negative three halves I square that I get positive nine fourths that's what it's added to both sides of your equation it's got to be both sides or else you don't equation okay it's got to be both sides this new one this one what's half that number everybody quickly please let's have that number one unless I want it's negative one they sign matters for the next thing we do it's negative when you square it what's the square negative one one so we're going to add that's always adding now we're going to add a to both sides can you do the last one please what's half that number come on okay here you go let's have it now well that's one half square one you're having one for people's remember the advice about a fraction calculator remember that refreshing fraction cap you know one of the place is not that hard but this sucks they have to think about all that stuff right there so with how much is it it's work four right so this 10:10 fourths that's five hats so 600 three plus one that's four did you guys get for it okay fraction got your buddy Calculon do it manual which time so all this junk has for this here's how you finish completing the square if you don't know how to do it it's really easy you just have to remember the numbers that you've got before you squared them and the factor is right there okay so here's how it works this part is going to give you something this part is going to give you something this part is commuter something this is X do you remember what happen ever was yet including the sign I remember now and square throughout what half that number was come on folks play along with me they wanna know maybe 1 minus 1 and you square this is no fun if I you can play tennis by yourselves come on substitute nepali get some like walls right now goodness gracious what's half of this what's hot button um the one that's got so z+ squared get one show Paintsville okay with that idea I know it's just algebra but that just algebra part can be really hard so that that's usually what people end up screwing up in calculus is the algebra and the graphing the doing this junk hey is that sphere can you recognize the formula for a sphere okay right now I want you to write we're going to write up here can you write the center and the radius for that sphere on your own right the center and the radius for that sphere please you don't have to say loud okay just go here true or false the first coordinate the x-coordinate of our center is going to be negative true or false I mean negative all good let's export a part of center reality it's always the opposite of that sign that's in there that's the lane is shifting words inside parentheses that's like a translation that you learn from innovating Alex wrote what's the Y coordinates are but what's the Z coordinate decoding perfect rays come on rate it's 416 - what is it / could you graph it if I asked you to yeah but not fun because even just finding the centre itself is going to suck and then doing the radius that that's not going to be so much fun computer yeah but you can think about where it's at and that's that makes you mean security at Northrop e on this one so far this last one that we're going to do just a little practice example to see how this stuff works I'm going to mostly just give you the the answers here I just want you thinking through it right now so let's think through this imagine a sphere and at opposite ends of the diameter I've got these two points a and B don't say a lot of once you think about it what would you do I'll just spit those nasty sorry cut that on HD camera whatever I'm not going to add it either enjoy that so how let's have trained the bot - oh yeah right these two points are opposite ends of a diameter diameters always go through where damage over the radius families go through the so if I give you two points say that the opposite end of I diameter how would you find the centre halfway through oh did we have a formula for the halfway through it's the distance the midpoint if you wanted to find out where the center is the center you should do this on your own when you go home to see if you the same thing the center would be at the midpoint of AD I've already done it the midpoint would be it's really just average you know I mean it's not that hard on so you cry you right now five hats negative one half levin's did you catch how to do the midpoint really funds to understand that the center would be the midpoint if that's the end of the day say get putting some in your head or I get an echidna it's a weird but there's a lot of challenger a lot you got a picture oh goodness could you tell me what the radius it could you tell me how to find the range could you tell me that okay I know that radius is half banana that's very good what's a diamond it's a segment from here to here how would you find that good yeah if you did the distance and then did the distance between here and here would sure give you the distance of the diameter how do you find the radius so this would be one-half a distance we need to keep the distance from A to B is it say 1 squared 5 squared 3 squared that's 35 in a square root and then take half that right there would be the radius square root 35 14 I did it fast you need to do it on your own where you know I'll just go through it just 31 times do the midpoint formula cell do the distance for himself take half of it to understand that the radius is going to be that square root of 35 or two or you guys are careful what I'm talking about how about this one if I give you the center which is what this is and the radius can you write for me the equation of the circle try it try it we just went from here to here try go from here to here on that example give you about a minute go for it write the equation until we get the good stuff get the favorite part of my day right here it wasn't when I got kicked out of my own classroom it wasn't when I ran my fifth wheel into an overpass it wasn't that something I was in today though it's still in my mind I'm so heartbroken just a little bit it happen for real and sucked it's when we could talk about vectors in like half a second when you finish the way that equate I think it's at certain limits fear by misspoke the equation of a sphere always looks like this always looks like that all you have to be able to do is figure out what the center is put it in the right spots it's kind of nice though because x coordinates match up with X variable y coordinates Y variables you know it's same stuff and the radius doesn't change either you can put the right things in the right spots the right signs and I can have a plus or a minus right here is very good for the square root of birth if I leave that square there yes and then if I actually go ahead and square it if you gave me this then that's all that's the correct answer Oscar you guys are think heart medium weird different right it's different get her up your head round you will trust me you will now we talk about vectors in 3-space this is where we make our money okay so this is this is great this is not calculus II stuff we use it to get our head around what 3d is this stuff this vectors in r3 in three space in three dimensions this is we're going to make our money in our class okay this is what we deal with calculus with is a lot of vectors so we're going to talk about vectors the good part is that all the stuff I'm teaching right now I'm really not teaching you I am just taking just like this was an extension of 2d this is an extension of 2d it's virtually the same okay so we talked about position vectors we're going to figure out what magnitude is we're gonna talk about unit vectors or kind of parallel we're not going to slope anymore because slope doesn't work anymore we don't we're not just on a plane we're crossing through planes and doing stuff like that so it doesn't work so they actually makes your lives easier there's not only going to be one way to check for parallel vectors are you following what I'm talking about so this if you think of anything it's an extension of 2d vectors but we'll talk about right now so the first thing we need to know is how we define position vectors let's you know that right come on once the position what's the position vector what's it what to do where does it start at zero or it starts at the origin 0 0 of 0 where does it go say it louder you set 2 to a point goes to a point so a position vector is this vector that has its initial point at the origin and it points to a point and where it point that point it points to are literally the components of that position vector it's very convenient it just says hey I'm going from the origin to here I'm pointing there and same thing in 2d we just now have four three the thing I need to know is that position vectors that's the idea is the same it starts it initiates at the origin and terminates at whatever the point is that's denoted by our vector Y understanding the concept here so it looks identical that would be old school position vector yes now here's no one this just a dizzy Pony and now instead of going on the XY plane we now have XY motion and Z motion gives us a 3d vector can you extrapolate from this and tell me maybe a different way to write without the brackets we think what would the first thing be bursting with Aayan first thing would be a number do you want it be a same exact number but then what this is is just different notations this is this is the vector that's pointing to this point do you get the point this one says that hey you're going this scalar times that I Direction need a vector in the X Direction then you're taking this in the J direction and you're taking well letter you want to use now guess genius is all of you okay we'll do K okay exactly right and what these are are the X and the y and the Z components X and the wine has eight components let's say it's the same thing as 2d which had that that extra little thing now if you remember two talking about the distance formula the distance formula was really similar to 2d which means the distance formula for vectors what do you call a distance formula for the distance of a vector a by the magnitudes we were two identical which is really nice so if you wanted to find the magnitude of a vector what do you think what do you think you're going to do come on what do you think you're going to do is we're square we're Adam come on do it it's wherever Adam Adam it's like people trade the dealer's find the supplier that it's pretty close it's like three things all right you're going to square really big parentheses no definitely do this we're going to square we're going to add them experiment it is the distance formula it just so happens that because position vector starts at zero zero we don't have that other you know this is also reserved so we're just squaring these numbers adding them and square root that's how it works okay the other two things and then literally all we have to do is practice are these final two statements which you already knew already basically no the first one because we don't have a slope concept in 3d right now in order to do that we have to restrict your direction and that's called directional derivatives and we're not there that's like chapter 13 okay so because we don't have a slope concept right now we can't talk about the vectors being parallel with their slopes the only other thing was the second way to do it from 2d do you remember it was like ten seconds ago instead of into it in 30 minutes that was a while back but do you remember it say what same unit vector or maybe slightly easier remember the thing we do is I write down here or you did the whole factoring thing come on from the other they were there was things that we were multiplying vectors by what are those called scalar that's how you tell you that's the that's how you tell we use the unit vector please please doesn't care pursue there's a slight difference we use the unit vector not to prove that vectors are parallel because that's pretty hard we use the unit vector to give us a parallel vector to something else that's what we use it for does that make sense shrink it and then grow it that's the unit vector doesn't shrink sit and we multiply the growing how do we check whether vectors are parallel is literally just if they're scalar multiples that's the only way that we really do in 3d so vectors are peptide that down please vectors are parallel if they are scalar multiples if you want the real Matthew way to write a is parallel to be fifth a is a scalar times B that's the way that this looks and we have done this already do you ever do it we took we factored out a number we said it's that number x times a or vice versa we've already done it so we're going to do that couple of times with redeemed that's basically it it is the math way of spelling it yeah it's the bike it will yeah it's it if and only if it's a by conditional so yeah I'm not I'm not that bad at spelling I know where is good yeah this is my conditional means that this is necessary and sufficient to prove this and this isn't it isn't necessary this one over there man so if this is good enough for this this is good enough for that so parallel means this is happening this means perilous happening that's this emergency can we practice you literally have enough I think honestly all of the rest of this is next stroke because it's just extrapolation but we're going to practice anyway so what we're going to do is maybe even like five examples we are going to fast through them because they're very similar to 2d are you guys okay with that okay so first thing I want to do is we're going to talk about parallel even though we just discussed it I'm going to show you how to how to prove it you let's prove that vector a this parallel to vector B when B is the thing that is gained yet let's let's prove that okay come on reiterate it for me I know we just talked about it I want you to get in your head here how would you show vectors are parallel what do they need to be a review parallel it was scalar that's easiest way to scalar multiples you can go down to the same unit vector but it's too much work okay that's a lot of work you typically do that to say hey I want a parallel vector parallel to another one then you've got to find the unit vector but if I just wanted to say if they're parallel had given you two of them well then let's risk factor factoring is finding a scalar multiple so let's inspect your can you factor that one what we did for you it are these two vectors parallel yeah are you getting good at finding the notation here and here and translating back and forth if you're not and I give you a different notation I'm going to do that a lot to get you used to it if you're not then you go one step further you okay this is I minus 2 J plus 5 K and this is 300 that's exactly right that's three times eight and then we go this is what I'm talking about if one vector is a scalar times another they are parallel with me yes no the minutes easy medium hard kind of easy to do right you just got to remember that this is what it is that's what the hard part just remembering all this stuff how to do it tell you what once you try this one down here's number two let's let B this vector I think this one's easier actually see it did you find out whether those two vectors are parallel or not what do you think don't let fractions concern you if you have a common denominator that's always a good thing to factor out almost always a good thing to factorize so if we have these thirds let's factor out the 1/3 then we get what you get thanks for whispering I still hear her G because I get good ears but yeah all right if i factor the 1/3 we get one on each guy minus 2 J plus 5 K if you get if you factor something and you get your vector you're checking for in there and you have a number times that you have something that is parallel these two things are also variable so fans if you look it with that interval okay so this is also don't this we're just going to work with what I want to do here is three things I'm going to make you do it on your own right now because I think that you can so I want to spend about two or three minutes I'm going to give you that time what I'd like you to do is find three things find thule - 3b hey can you take what you learn in 2d vectors and apply to that we can still multiply scalars by things we limit each didn't okay kind of undid multiplying we can add subtract vectors the same exact way so once you figure out what this is I want you to fear what this is tell what this is do you know what to do in each case noting that your if you do know what to do in each case good okay can you remind me to ask you a question like this one when we're done this one reminds I'll forget okay so we got like three more examples and then we're done with this section we just hammered have it through it so if you can hang on for me three more examples that we're going to go with our section here were you able to do we're able to do the first one they gave you enough time for this for the first one least okay we're able to do the all of them store okay let me give you another another minute or so I want you to actually do them all around I do want you to hang out with me to the last example on this because it's going to be a reiteration of the last example from section 11 point 1 and I want you guys to really hammer it into your head on how to do this okay so stick with me two last one oh goodness seven drivers you know a lot of times when we're switching formats or frankly just having a lot of small stuff going on in our head I don't know that I do that all at once in my head if you can't great that's fine I don't think I would go because I'd be like man that that's a lot of going on I got two different formats so pick one that works for you I don't care I don't care well I'll accept both answers is Jay's case or practice out here but be able to do it so for me personally be like okay let's do two a and see what it is I know it's negative 2i plus 4j that's my - a right here does that make sense then I'm going to do a minus signs I'm subtracting I'm just going to do three B well that's six well you can do the bracket notation you by effect brackets but that's 6i that's nice J and it's minus 3k it goes okay with that the first one if you did brackets that is absolutely fine I don't care are we done no but we've at least broken it down so let's go ahead and finish up the last this would be how much could you go back to that if you want to or negative 8i it doesn't matter what were anyone how about them up to J's what about the caves if you have negative 8 I - 5 j + 3 k that's that's exactly right also is this straightforward enough oil okay what's the first course of action on this what would you multiply what the real in case so for us this is this is literally just a 2d vector so when I'm finding this this was done last section we can do here but it's not to be a hard so we're going to have three negative 3i plus six j and then we're going to find magnitude of that square root 45 you're going to see a lot of non square root of 45 in the back of the book what do you see you see three or five because they're going to simplify that are you making the little mistakes yet I'd make it a little mistakes multiplying by the wrong number happens a lot happens to me that happens to me I'll be honest I do it all the time so that's why I take your time on these things so most ranges last one we find the magnitude of goodness thank you for it but you only that vector bracket notation that's fine negative 6 and positive 2 can take my work on that one did you all get the same thing and then find a magnitude I'm not going to do the work for you what's what's the magnitude of this how much root is it from 50 cent of it first room service to room 14 yeah this forward divides again I think as you remind about something here's my question if I took that negative way would it affect that all the negative does is reversed that direction does not affect the magnitude that's an industry right has no that's almost like the absolute value because it's a distance it's a magnitude to link this doesn't matter okay the last three examples we got we got these two one more than we're going to worry about good can you find and they're very fast so can you find a position vector between two points it's done exactly the same way that we found position vectors in Section 11 point one how did we do it I like the to teach but what's the main come on left ciders well how do you do it how do you do it how would you find a vector going from here to here and have a position vector do you remember if you do awesome if you're just kind of staring with fish eyes like oh crap don't make eye contact you call them I can call on you if we wanted that vector so this thing equals vector V it just subtracting just like it 2d it works in 3d so to find a position vector we do I'm going to bring it all out for you 1 - 2 4 - 1 5 - 0 position vectors no matter whether you're 2d or 3d are just about subtracting the coordinates of those points and then we automatically get a position vector is that clear enough for you guys subtract the courts the poison guy so here our vector would be negative three-five could you write that as a standard basis vector do you know what that means and I asked you to write as a standard basis vector if you do it yeah okay how old would it be you don't even do it just tell me how would it be written they divide 3j + 5 k yep that would be the other way to write that still okay could you find a could you find a unit vector for this if I asked you to that's a big deal in this class we're gonna find a ton of unit vectors can you find a unit vector here if I ask you to explain to me right side only how do you find a unit vector over the absolute can you say that one more time but without using apps of a search of jams breath United right search the M runs with Edna too good that's a joke not a funny one obviously goodness unit vectors are found the exact same way no matter what you were doing okay you take the original vector which I've given you divide by the magnitude go ahead and try that now would you please divide by the magnitude that's how unit vectors always work look if this is a length as your measure working through it if that's the length of your vector and you divide your vector by the length you're basically taking length divided by length the same length and getting 1 that's the definition of unit vector and I'll say this when I say unit vector what are we talking about okay doesn't like the one but what are you ready found directed direction we're finding the direction that this vector is headed in that's what we do okay let's focus on this one jib or have a second and we'll do that example and our section here today so our unit vector you've got this negative I plus 3 J minus K do I care if you switch that to the bracket notation if you prefer that do it but I do want to divide it by Eggman you gotta know how to find magnitude or sometimes at this point you got to know what that means and I think that you won't do so let's find the magnitude explain to me in the simplest way ever how you find the magnitude of vector what's the three things you do hollingers I've noted over but I'm imagining you said square I'm Adan's quickly to do so underneath the square root we're going to take just do the negatives matter we care it's just 1 1 9 1 I hope I did right yeah I end up squaring these in my head pretty much over time because I go okay negative 1 that's one switch 1 3 is 9 1 spiritual one I do that in my head a lot okay this is negative I plus 3 J minus K all over the square root of 11 this is not how you leave a unit vector you choose one of two ways from here on out one way is kind of sucky one ways a lot nicer you're gonna see both ways okay one way that you can write the unit vector make sure you have the little hats and unit vector you could rationalize and distribute all of this stuff and you get negative spirit 11 over a legend I plus 3 root 11 over 11 J minus square root of leavitt over 11 okay that's kind of a messy way to do it but it's the literal vector so there's pros and cons okay this is the vector gives you each direction independently and that's kind of nice in some circumstances we like this the other way to do it take this which is the thing that's taking your vector and shrinking it to a unit vector here take the thing rationalize it but leave it out in front of your vector and this is kind of more an interpretive in talents of interpretive dance it's like interpretive dance what's happening to the vector okay what's happening is you're taking this vector and multiplying it by something that's shrinking it this fraction is less than 1 its shortening that vector and that's it kind of interpret layout look at that that's what's happened to ends of universe it's happening good for you guys it's fantastic it doesn't matter who taken leave that like negative one no but does it have an application you do it you would okay last one I want you to recognize that this example is really similar last thing that we did in Section 11 point wine remember that give me a vector that is parallel I said a mess it up kind of give away the punchline here let's look at I'm answer questions at the very end here but let's focus up okay how won't you find the vector that has a clean semi or how do you think about 92 lengths of 2 magnitude legs impact and V as the 81 right now say direction means you know same what you know need a vector same direction means same unit vector this is when you have to find the unit vector here so you weren't talking about if you just check in whether two vectors are parallel skippable if I'm saying I want you to create from scratch a vector that is parallel the same direction as another one that's when you take this vector you shrink it you're vector and then you grow it by multiplying it's basically this but then the very end we're going to be okay you know what remember that one time when we took a magnitude and we like told me multiply it by a vector and we got this that's what we're doing so so basically when I asked you this problem on the test and I said once you find me a vector with that magnitude in the same direction as this what I'm asking you to do is find the direction that's why having you think unit vector stretch same direction same unit vector I want you to find me the unit vector and then give it the appropriate magnitude that's what I wanted find me the unit vector give it the appropriate magnitude Direction magnitude that's the vector is as a direction types of magnitude let me give you a minute I'm going to wrap it up you're really fast and we're going to call it good for this section question what are you first are they soreness I love that unit vector is that's the fame how do you find a unit vector that's like the big thing right you gotta find magnitude so when we do magnitude our unit vector is eatin too Thank You Man goodness you know what I see on tests a lot solve lie people take this and multiply by two and go yay you have found a parallel vector it's the same direction but it does not have the same magnitude does that make sense so we're going to run through a quick like the ideas you got short at first you got to find the unit vector for the direction thing multiplied by the two so in our case we got our I minus 2j plus 3k divided by let's see 9 squared 14 the magnitude of W is a square root 14 how I would write it and I'd seriously we do this I like this because it saves me a lot of heading and a lot of writing the same thing three times but you can do whatever you want you can distribute that and that's that's fine also what is this what is it what is it it's a unit vector what's universe mean come on unit vector means surveyor-general connection that's what it means you like the one it's the direction this is the direction that I want does that make sense but it has a magnitude of what no this has a magnitude of one on one because that is that unit vector let's give it the appropriate magnitude so right now we take our vector and okay the appropriate magnitude that I want was what pleased the magnitude sorry the the unit vector that I have was this one there's another good reason when you leave this hanging up front why lottie's you're simplifying and then if you have to then you can redistribute it if you if you have to reach me but the vector that we're talking about the vector that ends our section today which is nice don't do that I'll make little mistakes and ask the kiddies that's money you do that you get ten points on the test pretty-pretty means is it hard to do do you have to understand every little thing about it be good right though absolutely do there's a lot that's we're going to intersection today