Vector addition and subtraction
Adding & subtracting vectors
- [Voiceover] So let's get some practice and hopefully a little bit of intuition for adding and subtracting two dimensional vectors. So let's say that I have vector A, and let's say that its x component is I don't know, three, and it's y component is negative one. And let's say we also have a vector B. Vector B, and let's say that its x component is two, and let's say its y component is three. Now, let's first think about what would A plus B equal? So what resulting vector would this be? A plus B is equal to what? And I encourage you to pause the video and think about what this would be. Well, the convention is, is if we're taking the sum of two vectors, we can just add up their x, their x components to get our new x component and up their y components to get the new y component. So the x component of vector A plus vector B is going to be three plus two. That's going to be the x component, which we know is five. And the y component is going to be negative one. Negative one plus three. Negative one plus three, and so the resulting vector is going to have an x component of five and a y component of negative one plus three is equal to two. Alright, that was pretty straightforward. Now instead of adding them, let's think about what would happen if we subtract them. So let's think about what A, what vector A minus vector B would be, and I suspect that you might be able to guess or at least think about what would happen if you subtract vector B instead of adding vector B. Well, as you might have guessed, instead of adding the corresponding components, we subtract it. So our x component is going to be the x component of A minus the x component of B 'cause it's vector A minus vector B. So our x component is going to be three minus two. Three minus two, and our y component is going to be the y component of A, negative one, minus the y component of B. So minus three. Minus three, and so the resulting vector is going to be, is going to be the vector three minus two. The x component's going to be one, and then the y component of negative one minus three is negative four. Now what I just showed you, this is the convention for adding and subtracting two dimensional vectors like vectors A and B. Let's think a little bit about how we can visually depict what is going on. So let's first visually depict adding vector A and vector B. So let me draw some axes, and so my, so this could be, there's my y-axis. Y-axis, let's see. The highest y value I get to is, the highest y value I get to is three, and the lowest is negative four. And then, so let me draw the x-axis some place. Whoops, didn't mean to do that. I did the zoom by accident. Alright. So the x-axis, put it right over there. There's my x-axis, and let's see. The highest x value is five. The lowest is three. Actually, I could just focus on the, I could just focus on the, let me just focus on the right hand side. So that's y, and then let me do x looking like that. X-axis. And now let's see. Vector A is 3, -1. So one, two, three, and negative one. If we want to just visualize it in standard form, we could put its initial point at the origin and its terminal point at the point 3, -1. And so we could draw it like that. We could also shift it around, as long as it has the same, as long as it has the same magnitude and direction, we could shift it around. So that right there is our vector A. Now Vector B, 2,3, we could have its initial point at the origin and just, we could draw it like this. So the x-coordinate is two. Y-coordinate one, two, three. So its terminal point could be there. So we could just draw it like this for vector B, but if we're going to add vector B to vector A like we have right over there, what we want to do is shift vector B over so that its, I guess you could say, tail starts at A's head, or its initial point starts at A's terminal point. So let's do that. So if we start right over here, we're going to go two in the x direction. So we're going to go two more in the x direction, and we're going to go three in the y direction. So one, two, three. So we're going to end up right over there. So notice, this is the same vector, vector B. I've just shifted it over. Has the same magnitude and same direction as what I had drawn before, as that vector right over there. I have just shifted it down into the right. And now we do this, so this is vector A, I'm adding Vector B. I put the, I put the initial point of vector B at the terminal point of vector A. I've shifted it over right over there. So now I can figure out the resulting vector, A plus B, the resulting vector A plus B, by going from the initial point of A to the terminal point of B. So it's gonna start here, and then go over there, and notice, this vector I'm going five in the x direction, and two in the y direction. This is the vector 5,2 right over there. That is vector A plus vector B. But also think about what happens when we do A minus B. Well, we could still have vector A just like we drew it, but now instead of putting vector B's tail or initial point at the terminal point of vector A, we would wanna put negative B. So negative B would just be, have the same magnitude but it'll be in the opposite direction. So instead of going two to the right and three up, we would go two to the left and three down. So just two the left and one, two, three down, and we would end up right about there. And so, if you subtract A minus B, that's the same thing as A plus negative B. So negative B is going to look something like this. It's going to be the exact opposite direction. It's gonna look like that, and so there if you start at the initial point of A and get to the terminal point of the negative B now, this is all hand-drawn, so it could be a little bit more precise, notice we get to the point 1, -4. 1, -4. So this vector right over here, let me do that in a different color, this brown, whoops, that's not what I wanted to do. This brown vector right over here, that is the vector A minus B, and then this white one, actually, let me do this in another color as well, this magenta vector right here, that is A plus B. So hopefully that makes sense, and if you, given the components, you're just, if you're adding the vectors, add the correspondent components. If you're subtracting the vectors, well if you're subtracting B from A, subtract B's components, corresponding components from the corresponding components of A.