Subtracting vectors with parallelogram rule | Vectors | Precalculus | Khan Academy

In this video, we're going to think about what it means to subtract vectors, especially in the context of what we talked about as the parallelogram rule.

So, let's say we want to start with vector A, and from that, we want to subtract vector B. We have vectors A and B depicted here. What do you think this is going to be? What do you think is going to be the resulting vector? Pause this video and think about that.

All right, now the key thing to realize is A minus B is the same thing as vector A plus the negative of vector B. Now, what does the negative of vector B look like? Well, that's going to be a vector that has the exact same magnitude as vector B but just in the opposite direction. For example, this vector right over here would be the vector negative B.

Now, we just have to think about what is vector A plus the vector negative B. Well, there are two ways of thinking about that. I could put the tails of both of them at the same starting point—might as well do the origin. So let me draw negative B over here so we know the vector negative B looks like that.

One way that you are probably familiar with is you have vector A, and then what you do is you take a copy or you can think of shifting vector B so its tail starts at the head of vector A. If you did that, it would look like this. This is also the vector negative B, and then the sum of vector A and negative B (vector negative B) is going to be going from the tail of vector A to the head of vector negative B.

So, this would be the result right over here, which you could view as the sum of A plus negative B or the difference of vectors A and B—or vector A minus vector B.

Now, if we want to think about it in terms of the parallelogram rule, we could take another copy of vector A and put it so that its tail is at the head of this negative B, and then we would get it right over here, and we are forming the parallelogram.

Then the resulting vector is the diagonal of the parallelogram, and this just helps us appreciate that we could start with negative B and then add vector A to that or we could start with vector A and then add negative B to that. But either way, you get this white vector right over here, which we can view as the vector A minus vector B. And we're done.