Let's see if we can solve this example. So in this example, we are given vectors u, v, and w, and we're asked to sketch the result in vector for -2w+u-v. Now this is kind of an interesting problem because I noticed all the vectors that were given are directly horizontal or vertical. But there is some good news with this example, and that's we can use the same methods for the tip to tail method and the vector algebra we've already learned about to solve this problem as well. So let's just see how we can do this. Now, whenever we're dealing with these types of multiple vector operations like we have here, I like to start with one piece of this and then keep working on different pieces until we connect everything together and get our final vector. Now the piece I'm going to start with is this u-v, because I see this is within the parenthesis. Now I can see vector u is right about here, and I can see vector v is right there. Now vector v would this would be vector u+v if we were to connect these tip to tail. We're looking for u-v, and to find u-v, I need to make v negative. So v currently points to the right and is positive, and to make this negative, I need to have it point in the opposite direction. It'll keep the same magnitude, but it will point in the opposite direction. So notice it's now pointing to the left, that is vector negative v. So if this is vector negative v, what I can now do is find vector u-v by connecting vectors u and negative v tip to tail. So we're going to have negative v, where we have the tail of negative v drawn at the tip of vector u, and then I'll connect these vectors with a resultant vector which looks like this. So we're going to have the initial point of the first vector and the terminal point of the last vector, and connecting those two points will give us the vector u-v. So now that we found this vector u-v, I'm going to next focus on the vector -2w. Now I can see that we have vector w right here it's 1, 2, 3, 4 units to the left. And if I want to find 2w, that's going to be twice the length of w. So if we go 4 units to the left, vector 2w is going to go 8 units to the left. So we're gonna go 1, 2, 3, 4, 5, 6, 7, 8 units to the left, and that right there is going to be vector 2w. But we're not actually trying to find 2w, we're trying to find -2w, and negative 2w is going to have this vector pointing in the opposite direction. So since this is vector 2w, vector negative 2w is going to start right here, and it's going to have the same length but will point in the opposite direction. So this would be vector negative 2w. Now all I need to do at this point is connect these vectors together because we found vector negative 2w, and we found vector u-v. So to find -2w+u-v, I can use the tip to tail method. So the way that I'm going to do this is I'll actually take this negative 2w vector that I just found, I'm going to redraw it so it's actually up here. So it's actually going to go from this point all the way over there, that's negative 2w. And then we have the tip of vector negative 2w drawn at the tail of vector u-v. So all I need to do is draw this resultant vector to solve the problem, and this would give us the vector -2w+u-v, which we have right here. So this right here is the vector that we're looking for, and that is the solution to this problem. So, I hope you found this video helpful. Thanks for watching.
8. Vectors
Geometric Vectors
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