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Parallelogram rule for vector addition

The parallelogram rule says that if we place two vectors so they have the same initial point, and then complete the vectors into a parallelogram, then the sum of the vectors is the directed diagonal that starts at the same point as the vectors. Created by Sal Khan.

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Video transcript

- [Instructor] So we have two vectors here, vector A and vector B. And what we're gonna do in this video is think about what it means to add vectors. So for example, how could we think about what does it mean to take vector A and add to that vector B. And as we'll see, we'll get another third vector. And there's two ways that we can think about this visually. One way is to say, all right, if we want start with vector A and then add vector B to it, what we can do, let me take a copy of vector B and put its tail right at the head of vector A. Notice I have not changed the magnitude or the direction of vector B. If I did, I would actually be changing the vector. And when I do it like that, this defines a third vector which can be use the sum of a plus B. And the sum is going to start at the tail of vector A and end at the head of vector B here. So let me draw that. So it would look something like that. And we can call this right over here, vector C. So we could say A plus B is equal to vector C. Now we could have also thought about it the other way around. We could have said, let's start with vector B and then add vector A to that. So I'll start with the tail of vector B and then at the head of vector B, I'm going to put the tail of vector A. So it could look something like that. And then once again, the sum is going to have its tail at our starting point here and its head at our finishing point. Now, another way of thinking about it is we've just constructed a parallelogram with these two vectors by putting both of their tails together, and then by taking a copy of each of them and putting that copy's tail at the head of the other vector, you construct a parallelogram like this, and then the sum is going to be the diagonal of the parallelogram. But hopefully you appreciate this is the same exact idea. If you just add by putting the head to tail of the two vectors and you construct a triangle, the parallelogram just helps us appreciate that you can start with the yellow vector and then the blue vector or the blue vector first and then the yellow vector. But either way, the sum is going to be this vector C.