last video we saw how to represent and do math with vectors graphically and now we're going to segue over to how to do it with components and the bridge between these two is our special vectors known as unit vectors now a unit vector is literally that a vector of length one now in physics we're always very careful to say one what how many you know what what physical units do you have associated with that and the answer here is there are no units this is just basically the number one given a direction and you can make these things point anywhere you want but in but for what we want right now there are three special ones that point along the x y and z coordinate axes and so these get called i hat j hat and k hat and these are vectors of length 1 that point in the x y and z directions accordingly now you might ask why don't you call them x hat y hat and z hat and in fact some people do um this is more for legacy reasons because it used to be that in order to represent positions in 3d before the vectors were invented mathematicians and physicists used an extension of complex numbers called quaternions and so just like i is the square root of negative 1. in this extension j and k were also equal to square root of negative 1 but not equal to each other but the product of them was those details don't really matter but suffice it to say that the i j and k is a holdover from the days when position when what we would now use vectors were represented using this extension of the complex numbers known as quaternions and you'll notice that although these are vectors i didn't write the little half arrow vector signy thingy instead i wrote a circumflex which usually you just read as hat because it looks kind of like a little hat and this is a special notation for unit vectors so these are vectors of length one with no physical dimension length one no physical dimension and um in this series of videos we're not going to stress too much about k hat which points in the z direction because most of the time we can keep our motion confined to a plane and so be mostly interested in the x and y directions so i hat and j hat are going to be the ones that we're most interested in and the idea with these unit vectors is that we can construct any vector we want by taking these unit vectors and applying the rules of a scalar multiplication we multiply these unit vectors by whatever scalar quantities we want and then we can add that together to construct any vector we want so let's say that um i have a velocity like so and we'll say that's magnitude is 5 meters per second i'm choosing that one very specifically let's say this is angled 37 degrees above the x-axis i can construct this by first taking my vector i hat which points to the right or our points east and multiply it by four meters per second like so so this vector here is i hat the thing that points right that i've times by 4 meters per second so i've given it its physical dimension now when i've multiplied it by my scalar quantity now i can add to it here i'll do it by tail to head on a vector j hat sorry three three meters per second times my vector j hat so j hat points up or north um i multiplied this by three meters per second and if i want to point south i would multiply by negative three meters per second in any event i can now write my velocity vector as 4 meters per second times i hat plus three meters per second times j hat and for completeness sake if you want we can say plus zero meters per second k hat but i'm not too worried about that all right with this um we can start to segue into component vectors which are not the same thing as vector components but we will tie these together very quickly um so we sort of already introduced the idea without necessarily realizing it um by constructing a quantity out of unit vectors so let's say this is my vector a like so um i can following the same sort of idea and here i'll use the parallelogram rule i can imagine constructing my vector a by adding this special vector that points purely in the x direction and this special vector that points purely in the y direction this is a sub x and a sub y with vectors on them and you can see i can complete the parallelogram to make my vector a so i can say that a x plus a y equals my vector a all right and again just as a reminder the magnitude of my vector a we would just write as a and this is always a positive number because magnitude just means the length of the vector so these vectors here go by especially if component vectors as in these are the components you need to add this up now this is not the same thing as the sim as the related and confusingly named concept vector components which you'll use much more compound vectors are sort of bridged to that so the idea here is i if we go back from before since i can get any vector by timesing it by in a scalar by i hat j hat or k hat i can get any compound vector doing that i have points along x so i can make any ax equal to some number times i hat similarly any a y is some number times j hat so this this number that you need is the vector component and you just write it as a x or a y without the vector sign so a x a y are what you need to multiply oops let's make that look a little better small to apply i-hat and j-hat by to get a x and a-y the vectors so we call these the x and y components of the vector a and a lot of times you just skip through the whole component vector thing and just diagram in the components directly so say this is my vector a i would just say drop down a horizontal and a vertical like this i would just write a x and a y like that and we would say that this is the x component of a and this is the y component of a and importantly here these quantities here this can be positive or negative if it's positive it's going to the right or east or whatever and negative would be left or west or whatever and similarly this could be positive or negative this would be up or down or north and south so even though we write this like it was the magnitude of a x and a y that isn't quite right it's the number you need to multiply i hat or j hat by to get the component vector confused the notation is maybe just a shade confusing but you get used to it pretty quickly all right so let's say you have the magnitude and direction of a vector and you want to find the components how would you do that um you do trig so just as a quick example here let's say this is my vector b i'm going to have it point down into the right like that and we'll say that this angle theta here is 37 degrees again why 37 degrees um technically it's 36.9 but 37 degrees and 53 degrees are the interior angles of a 3 4 5 right triangle so that way i can do the trig in my head so let's say i want to find the x and y components of my vector b again i can just imagine extending out the legs of the right triangle here so this would be b x this would be b y and the idea here and let's say my vector b is 10 meters long um so the idea here is i just use um trig so we'll just go with a good old sohcahtoa if you haven't seen this mnemonic before this is just a way to remember how sine cosine and tangent work so the sine of theta that's the s is the opposite over the hypotenuse c for cosine is the adjacent over the hypotenuse the t in toa is for tangent theta and that's opposite over adjacent o over a and you'll have to put in your plus and minus signs by hand but let's go ahead and first and think about trying to find bx so i know the hypotenuse i want the adjacent so that's going to be cosine here so we can say cosine theta is the adjacent over the hypotenuse oops so that's b x over b so b x is equal to b cosine theta so that's 10 meters times the cosine of 37 degrees is plus 8 meters and the reason is plus 8 is because the vector b is pointing in a right-ish direction all right now what if i want so now let's go ahead and find b y now here i'll have to put in a minus sign by hand but we can play the same game this is the opposite so we can say that the sine of theta is the opposite over the hypotenuse um so this would be b y over b and i'll stick in a minus sign here because i know that um b y should be a negative number but when we do trig and physics we usually try to rig it up so that all of our angles are between zero and ninety degrees so mathematically they're in quadrant one even though our vector b here clearly points into quadrant four the reason that we like mathematically making everything quadrant one is because of how calculators work while the regular trig functions are just fine inverse trig functions in particular have to make choices about what quadrant the answer is in and different calculator manufacturers make different choices about what quadrant things point in so the only quadrant that you can trust for sure in a calculator is quadrant one so physicists like to make all of their angles to be between 0 and 90 degrees just to keep everything in quadrant 1 in your calculator which means that we have to take care of the plus and minus signs ourselves since we know this is really pointing in quadrant 4 and b y should be negative that's why i had to put the minus sign there because i'm mathematically forcing my theta into quadrant one so b y here will be equal to minus b sine theta so so this will be minus 10 meters sine of 37 degrees will be negative 6 meters and i interpret the minus sign as meaning we're going down okay well let's do the reverse example um so let me just copy paste this real quick copy oops paste there we go so let's say we already knew our vector components but we wanted to find the magnitude and the direction so we'll use the exact same values we had from before so we know what the answer is going to be but let's just work it through so let's say bx is 8 meters b y is negative 6 meters i want to know what is the magnitude of b and what is this ang direction angle here so i can get the magnitude here by the pythagorean theorem so b squared since that's hypotenuse will be equal to b x squared plus b y squared since these are the legs of a right triangle now you take the square root of both sides get that b is the square root of b x squared plus b y squared like that so b will be the square root of 8 meters square quantity squared plus negative 6 meters quantity squared all right so when i square things the minus sign goes away both the number and the unit gets squared which is good because this is going to give me like 64 square meters here this will give me 36 square meters here for a total of 100 square meters now the important thing is when i take the square root i have to take the square root of both the 100 and the square meters so leave me with 10 as the square root of 100 but then the square root of square meters is just meters all right so we got that the magnitude is 10 meters now the way we get the direction is via the arc tangent trick so we can start by saying that the tangent of theta is the opposite over the adjacent whoa let's try that again opposite over adjacent there we go um and again remember this is really living in quadrant four but mathematically we want to force everything into quadrant one to make sure that our calculators love us so we're just going to solve for this and then at the end i'll show you how we indicate what quadrant we're in so i need to force this to be the opposite to be a positive number and i know it's negative so i'm going to say minus b y bx is positive so it doesn't need any forcing all right now i'll take the arc tangent of both sides so on the left the arc tangent of the tangent theta is just theta itself and on the right now we have the arc tangent of minus b y over b x again what we're doing here is we're mathematically forcing this into quadrant one to make our calculator like us so this is going to give me the arc tangent of minus a minus six meters over eight meters now let's look at a couple things here first first off i have minus minus is a plus that's good so now i'll be taking the arc tangent of a positive number and see this is the thing is that arc tangent positive number could return a uh an angle in either quadrant one or in quadrant three your calculator doesn't know which so calculator manufacturers have to just assume a quadrant and they they do all assume quadrant one there are similar issues with arc sine and arc cosine and calculator manufacturers do not consistently make they do not make consistent choices of quadrants for negative inputs into arc sine arc cosine or even arctangent this is why we force everything into quadrant one second thing is notice that the meters here cancel and this is super important whenever you are taking an inverse trigonometric function it has to be dimensionless i don't know what an arc tangent of meter means so any physical units you have going into an inverse trig function always need to cancel if they don't that means you made some sort of a math mistake somewhere all right punch all that through your calculator you get 37 degrees well that's awesome but 37 degrees where so by convention what we'll do is we'll say if this direction is east b is heading south of east and so we would finish this off by saying 37 degrees south of east like that we could also say that it is 53 degrees east of south that is a perfectly valid description alrighty so the this now will lead us into doing vector arithmetic using components so the idea goes like this we know that we can construct any vector by adding up its component vectors and we can construct any component vector by taking unit vectors and multiplying them by the appropriate scalars so the what mathematicians realized is that we don't even really need to draw the arrows anymore so let's say i'm adding two displacements we'll make my first displacement um negative two comma one meters now what this means is that this vector is the same difference as adding um negative 2 meters i hat plus 1 meter times j hat but here i'm not even bothering to actually write the i hats and j hats anymore um i'm just writing a list of numbers and mathematicians have shown that you can represent a vector purely as a list of numbers and that's really cool all right let's uh go ahead and pick a second displacement so this is four comma three meters and again same idea that's the same difference as writing 4 meters times i hat plus 3 meters times j hat alrighty now a quick word about notation you'll notice that i wrote the displacements delta r in previous videos i wrote things like delta x and delta y what's up with that and that's because of a unique annoying bit of physics notation having to do specifically with positions and displacements the name of the position vector in physics is r with a little arrow over it and since the displacement's a change in position the name of a displacement is delta r now the names of the components you would think would be r sub x and r sub y but we write those so often that we just write x and y so that this if motions purely along the x direction that's why we wrote delta x is that it's technically delta r sub x but we never bother actually um writing out the subscript and this is a thing we do specifically for the vector r our position vector and delta r our displacement vector for anything else like a velocity the name of the velocity vector is v its x component is v sub x its y component is v sub y acceleration is a it's x components a sub x it's y components a sub y but we don't write r sub x and r sub y if you do people would probably know what you're doing but nobody writes it that way so anyway let's say that our total our journey was made of two different um uh phases so we want to know the displacement for the whole journey delta r i can do that by adding up the two displacements because after all we don't care about the path we just care about how we ended up and so here's the trick is what you can do is you can just add these slot wise so you say negative two comma one meters like that plus four comma three meters like that so you just go ahead oh sorry negative yeah negative 4 count of 3. so you just go ahead and you add the x components together so negative 2 plus 4 is plus 2 and 1 plus 3 is 4. so our net displacement is 2 meters east and 4 meters north and that makes sense if we take a look at this um we were saying we went first two meters to the left and then we ultimately ended up four meters to the right of where we started from so up two sorry four meters to the right of that so we ended up two meters to the right of where we started from and we went one meter north and three meters north to end up a grand total of four meters north of where we started from so cool so we can just write the numbers and add them up in slots and vector subtraction works the same kind of way let's say i have an initial and a final position so we'll say my initial position is three comma two meters and my final position is four comma six meters i can do the subtraction slot wise as well so my delta r would be our final minus r initial because remember changes are always final minus initial so i just go ahead and arrange it in slots like this and subtract each slot individually four minus three is one six minus two is four so i have my displacement vector is one comma four meters so that means that the position four comma six is one meter to the right and four meters above my initial position and again this doesn't tell me anything about the actual path i took to get there it just says how i ended up all right so how do i go ahead and do something like a multiplication by a scalar um so let's just as an example here oops let's say i have a velocity um so this is negative seven comma nine meters per second so this piece would be my v sub x and this piece would be my v sub y and let's say i say that something is going twice as fast as that so what is its velocity um the idea of something like this is you distribute the scalar across the vector components so 2 times negative seven will be negative fourteen two times nine will be eighteen meters per second like that all right and because i didn't really know where else to stick it one final little bit of notation um some physicists are not terribly hung up about x and y being horizontal and vertical it'll be whatever is convenient for a problem so a lot of problems will be seen in the not too distant future will involve things like say a crate sliding down a ramp or something like that with these so-called incline plane problems um it is almost always the case that you want to choose a coordinate system that is where the x-axis is parallel to the inclined plane like that so this would be my x and my y coordinate like that um we'll get in um this is because the the object will be accelerating down the ramp it'll be mathematically convenient to make the acceleration point in just one axis but that's a coming attraction but let's say that we've got a vector a in our coordinate system here if i take the components of it um again it's the same deal but now you gotta be really careful like to get my ax here i'm doing is i'm dropping a perpendicular down to the x axis like that and this will be my a x right there and similarly to get a sub y i need to drop a perpendicular over to my y axis and this piece right here will be my a sub y there let's make those look the same now for this kind of a thing a lot of times when direct there's a natural thing that you're trying to align axes to be parallel or perpendicular to instead of writing this as a sub x a lot of times we'll write this as a parallel instead of writing this as a sub y we will write this as a perpendicular all righty so in the uh next set of videos we will be taking this idea and expanding it out to doing uh kinematics in more than one dimension so we will catch you over there