In this example, we are given vectors u, v, and w, and we're asked to sketch the resultant vector u + v - w. So let's see how we can do this. Now, what I'm first going to do is rewrite this operation over here. So we're going to have u, but rather than having v - w in the parentheses, I'm going to write this as v + (-w). This will just allow us to see what our vectors are going to look like on this grid.
Now, I'm first going to deal with what's inside the parentheses and I'm going to find -w. Now, I can see that vector w is right there and if I negate w, it's going to have the same magnitude but opposite direction. So, we're going to start here and we're going to finish over there because it's going to be pointing in the opposite direction as w. This would be vector -w.
So now that we found -w, the next thing I'm going to deal with is finding v + (-w). And I can do this using the tip-to-tail method. So, I can see that the tip of vector v is right there and the tail of vector -w is right here. So if I go ahead and move -w up there, I can connect these tip to tail and I can see that vector -w is 1, 2, 3, 4, 5, 6 units to the left. So, starting here, we're going to go 1, 2, 3, 4, 5, 6 units to the left and this right here is vector -w.
Now, to find vector v + (-w), I can just draw the resultant vector. The resultant vector will start here and finish there, and that is going to be the vector v + (-w) which is also v - w; it's okay to write it like that as well. So this is what we end up getting. Now, from here, what I need to do is find vector u + v - w. We already figured out this is vector v - w, so what I'm going to do is use the tip-to-tail method on this vector. So, we'll have the tip of vector u connected to the tail of vector v - w.
So I'll write the v - w vector right there, and I see that this vector is 1, 2, 3 units to the left and 2 units down, so we'll go 1, 2, 3 units to the left and 2 units down, and that's going to be vector v - w and I'll put this in parentheses just to match what we have in the problem. And then all I need to do is draw another resultant vector which is going to go from the initial point of u to the final point or terminal point of v - w, and that's going to give me the vector u + v - w, and that right there is the vector and the solution to this problem.
So, I hope you found this video helpful. Thanks for watching.