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Combined vector operations

Watch Sal find new vector 3u + 1/5w when u = (2, -1) and w = (-5, 5).

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Video transcript

- [Voiceover] So what we have right over here, we have the vector U and we've defined it by giving its X & Y components it's a two-dimensional vector, and we have the vector W. And we've graphed them, the vector U, its X component is two, its Y component is negative one. So if we put its initial point at the origin, the terminal point, or its head, will be at the point two, comma, negative one, which is right over there. And for vector W, it's negative five, comma, five. So its X component is negative five. So if we start at the origin we would move five to the left. And its Y component is positive five, so we would move five up then to get to the head of the vector, or to get to the terminal point. Now given these two vectors, what we wanna do is evaluate what three times the vector U, plus one fifth times the vector W is, and I encourage you to pause the video and to give a go at it. Well three and one fifth are scalers. They are going to scale these vectors and we're gonna see them, see that happen visually. So we're gonna scale up vector U by three, we're gonna scale down vector W, we're gonna multiply it by one fifth, and then we're gonna add the resulting vectors. So let's do that. So when we scale vector U by three, we could just do this as three times vector U, we know is a vector two, comma, negative one, and so that's going to be, we could write it this way, so let me write the two, let me write the negative one, that's going to be three times two, for the new X component once we scale it up. And three times negative one for the new Y component, and of course that's going to be, result in the vector three times two, our new X component is six, and our new Y component is going to be negative three. And so let's plot that. So everything I've done just now is that part of the expression, that part of the expression. So the vector six, comma, negative three, if we start it at the origin, we're going to move, let's see we're going to move one, two, three, four, five, six in the X direction and negative three in the Y direction. So one, two, three, one, two, three, so we would get right about there. So there you have it, this is the vector three U. And we're at one, two, three, four, five, six in the X direction, or six to the right, and then we went down three, one, two, three, in the Y direction, and notice, it's in the exact same direction as vector U, it just has three times the magnitude that's vector U, that would be two U. And then we get to three U, three times the magnitude. Alright, now let's figure out what one fifth times W is. Let's see one fifth times W, well let me just write this, it's gonna be plus one fifth, W is the vector, is the vector negative five, comma, five, and so this is gonna be plus, so that's gonna be, let me write the components down. So it's gonna be plus one fifth, times, one fifth times negative five, and the Y component is gonna be one fifth times five, and so that's gonna be plus, whoops I wrote one half, my brains not working properly, one fifth times negative five, and one fifth times five. And so, this part right over here is going to be one fifth times negative five, is negative one, and one fifth times five is positive one. And so this new vector, one fifth W, is gonna be W scaled down. And so it's negative one, comma, one if we start it at the origin, so negative one, comma, one. We'd get right over there, and notice, it's going in the same direction as W, it's just one fifth, it's just one fifth as long. And now we just wanna add these two vectors. So if we add them by just looking at its components, the resulting vector, the resulting vector, let me do this in a new color that I have not used yet, so the resulting vector, we're gonna add the corresponding X components, so it's going to be six plus negative one, six plus negative one, and the resulting Y component is going to be negative three plus one. Negative three plus one. And so the resulting vector is going to be equal to five, comma, negative two. And we can also see that visually. If we start with this blue vector, three times the vector U, and we were to add the green vector, one fifth W, well we were to add that we would just start at the head of three U, then we're gonna add negative one, comma, one, so we're gonna move one to the left and one up, we're gonna get right over there. So let me see if I can draw that. So, just gonna do a little head to tail method right over here, so the head of the first vector is gonna be where the tail of the next vector starts, that we're adding. And so the resulting vector is going to be, if we started at the tail of the first vector, three U right over here, or at the origin, and then we bring it to the head of the second vector, we get, once again, the vector five, comma, negative two. Its X component is five, we're gonna move five to the right, and we move two down.