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# Visualizing vectors in 2 dimensions

## Video transcript

all the problems we've been dealing with so far have had have essentially been happening in one dimension you could go forward or back so you could go forward or back or right or left or you could go up or down what I want to start to talk about in this video is what happens when we extend that to two dimensions or we can even just extend what we're doing this video to three or four really an arbitrary number of dimensions although if you're dealing with classical mechanics you normally don't have to go more than three dimensions and if you're going to deal with more than one dimension especially in two dimensions we're also going to be dealing with two-dimensional vectors and I just want to make sure through this video that we understand at least the basics of two dimensional vectors remember a vector or something that has both magnitude and direction so the first thing I want to do is just give you a visual understanding of how vectors in two dimensions would add so let's say I have a vector right here that is vector a so once again its magnitude is specified by the length of this arrow and it's Direction is specified by the direction of the arrow so it's going in that direction now let's say I have another vector let's call it vector V let's call it vector B it looks like this and what I want to do in this video is think about what happens when I add vector a to vector B so there's a couple things to think about when you visually depict vectors the important thing is for example for vector a that you get the length right and you get the direction right where you actually draw it doesn't matter so this could be vector a this could also be vector a notice it has the same length and it has the same direction this is also vector a I could draw a vector a up here it does not matter I could draw the vector a up there I could draw a vector V I could draw a vector B over here it's still vector B it still has the same magnitude and the direction notice we're not saying that its tail has to start at the same place that vector a's tail starts at i could draw a vector b over here so i can i can always have the same vector but i can shift it around so i could move it up there as long as it has the same magnitude the same length and the same direction and the whole reason I'm doing that is because the way to visually add vector if I wanted to add vector a if I wanted to add vector A plus vector B plus vector B and I'll show you how to do it more analytically in a future video I can literally draw a vector a I draw a vector a so that's vector a right over there and then I can draw a vector B but I put the tail of vector B to the head of vector a so I shift vector B over so its tail is right at the head of vector a and then vector B would look something like this it would look something like this and then if you go from the tail of a all the way to the head of B all the way to the head of B and you call that vector C and you call that vector C that is the sum of a and B that is the sum of a and B and it shouldn't make sense if you think about it if these were let's say these were displacement vectors so a shows that you're being displaced this much in this direction B shows that you're being displaced this much in this direction so the length of B in that direction and I were to say you have a displacement of a and then you have a displacement of B what is your total displacement so you would have had to be I get shifted this far in this direction and then you would be shifted this far in this direction so you the net amount that you've been shifted is this far in that direction so that's why this would be the sum of those now we can use that same idea to break down any vector in two dimensions into its we could say into its components and I'll give you a better sense of what that means in a second so if I have vector a let me pick a new letter let's call let's call this let's call this vector vector X let's call this vector X I can say that vector X is going to be the sum of this is going to be the sum of this vector right here in green and this vector right here in red notice if I take if I X starts at the tail of the green vector and goes all the way to the head of the magenta of a correct vector and the magenta vector starts at the head of the green and then and then finishes I guess well where it finishes is where vector X finishes and the reason why I do this and you know this hopefully from this comp explanation right here says okay look vector Gris the green vector plus the magenta vector gives us this X vector that should make sense I put the I put the head of the green vector to the tail of this magenta vector right over here but the whole reason why I did this is if I can express X as the sum of these two vectors it then breaks down X into its vertical component and its horizontal component so I could call this I could call this the I could call this the horizontal component or I should say the vertical component X vertical and then I could call this over here I could call this over here the X horizontal or another way I could draw it I could shift this X vertical over remember it doesn't matter where I draw it as long as it's the same magnitude and direction and I could draw it like this X vertical and so what you see is is that you can express this vector X you can express this vector X let me do the same colors you can express this vector X as the sum of its horizontal and its vertical components as the sum of its horizontal and its vertical components now we're going to see over and over again that this is super this is super powerful because what it can do is it can turn a two-dimensional problem into two separate one-dimensional problems one acting in a horizontal direction one acting in a vertical direction now let's do it a little bit more mathematical I've just been telling you about length and all of that but let's actually break down let me just show you what this means to break down the components of a vector so let's say let's say that I have a vector that looks like this let me do my best to so let's say I have a vector that looks like this its length is 5 so let me call this vector a and I will say so vector A's length is equal to 5 and let's say that its direction we're going to give it we're going to give us direction by the angle between the direction it's pointing it and the positive x-axis so maybe I'll draw some an axis over here so let's say that this right over here is the positive y-axis going in the vertical direction this right over here is the positive x-axis going in the horizontal direction and to specify this vectors direction I will give this angle right over here and I'm going to give a very peculiar anger but angle but I picked this for specific reasons just so things work out neatly in the end and I'm going to give it in degrees it's thirty six point eight six nine nine degrees so I'm picking that particular number for a particular reason and what I want to do is I want to figure out this vectors horizontal and vertical component so I want to break it down into something that's going straight up or down and something that's going straight right or left so how do I do this well one I could just draw them visually see what they look like so it's vertical component would look like this so let me it's it would start it would its vertical component would look like this and it's horizontal component would look like this its horizontal component would look like this the horizontal component the way I drew it would start where vector a starts and go as far in the X direction as vector a is a tip but only in the X direction and then you need it to get back to the head of vector a you need to have its vertical component and we can sometimes call this we could call the vertical component over here a sub y just so that it's moving in the Y direction and we could call this horizontal component a a sub X and what I want to do is I want to figure out the magnitude of a sub y and a sub X so how do we do that well the way we drew this I've essentially set up a right triangle for us this is a right triangle we know the length of this triangle the length of or the length of this side or the length of the hypotenuse that's going to be the magnitude of vector a and so the magnitude of vector a is equal to five we already knew that up here so how do we figure out these sides well we could use a little bit of basic trigonometry if we know the angle and we know the hypotenuse how do we figure out the opposite side to the angle so this right here right here is the opposite side to the angle and if we forgot some of our basic trigonometry we can relearn it right now so uh so katoa sine is opposite over hypotenuse cosine is adjacent over hypotenuse tangent is opposite over adjacent so we we have the angle we want the opposite and we have the hypotenuse so we could say we could say that the sine the sign of our angle the sine of thirty six point eight nine nine degrees is going to be equal to the opposite over the hypotenuse the opposite side of the angle is our what is the magnitude of our Y component is going to be equal to the magnitude of our Y component the magnitude of our Y component over the magnitude of the hypotenuse over this length over here which we know is going to be equal to five or if you multiply both sides by 5 you get 5 sine of thirty six point eight nine nine degrees is equal to the magnitude is equal to the vertical component the magnitude of the vertical component of our vector a now before I take out the calculator and figure out what this is let me do the same thing for the for the horizontal component over here we know this side is adjacent to the angle and we know the hypotenuse and so cosine deals with adjacent and hypotenuse so we know that the cosine of thirty six point eight nine nine degrees is equal to cosine is adjacent over hypotenuse is equal to the magnitude of our X of our X component over the hypotenuse the hypotenuse here has or the magnitude of the hypotenuse I should say which has a length of five once again we multiply both sides by five and we get five times the cosine of thirty six point eight nine nine degrees is equal to the magnitude of our X of our X component so let's figure out what these are let me get the calculator out let me get the let me get my trusty ti-85 out one make sure it's in degree mode so let me check yep we're in degree mode right over there I want to make sure we're not in Radian mode now let's exit that and we have the vertical component is equal to five times the sine the sine of thirty six point eight nine nine degrees which is if we round it right at about three so this is equal to so the magnitude of our vertical component is equal to three is equal to three and then let's do the same thing for our horizontal component for our horizontal component so now we have five times five times the cosine of thirty six point eight nine nine degrees is if we once again if we round it to I guess our hundreds place we get it to being four so we get it to being four so we see here is a situation where we have this is this is a classic three four five Pythagorean triangle the magnitude of our horizontal component is for the magnitude of our vertical component the vadik the magnitude of our vertical component right over here is equal to three and once again you might say Sal you know why why are we going through all this trouble and we'll see in the next videos if we say something has a velocity in this direction of five meters per second we can actually say that we can break that down into two component velocities we can say that that's going in the upwards direction at three meters per second and it's also going to the right in the horizontal direction at four meters per second and it allows us to break up the problem into two simpler problems into two one-dimensional problems instead of a bigger two-dimensional one