 So now we can see how to add vectors using a graphical technique. As we move through physics, there's going to be several different applications where we need to add vectors up for various reasons. First one we're going to start with is the concept of motion along a series of paths. For example, you move along path A, then along path B, etc. In the end, you want to find your final position. Where did you end up after you did all these little paths? Well, the graphical method, sometimes called the tail-to-head method, involves tracking along your first vector A, and then from the head of that one, you move to the tail of your second vector along path B. The addition of those two vectors ends up being the side of the triangle here, or connecting from the first tail to the last head. Let's take a look at some examples of this now. Now we can do some examples of using this head-to-tail vector addition. I've got two vectors here, A and B. If I want to add vector B to vector A, graphically, all I need to do is place the tail of vector B on the head of vector A. And now I've gone along path A, followed by going along path B. The end result of this, or what we sometimes call the R vector, is A plus B equals R, going from the first tail to the last head. One of the beauties of the head-to-tail method, though, is I can use it for more than just two vectors. If instead of having just A plus B, I added in a third vector. Let's call it C. And then took a fourth vector, D, on and on and on as much as I can. Go along path A, then along path B, then along path C, then along path D. The result is a new position defined by moving from the very first tail to the very last head. This is our new result vector of the addition of A plus B plus C plus D. So here's another example where I can take an A vector plus a B vector and find the result of that using our tip-to-tail method. If I use this method, though, I'm going to show you something else. Let's say I had two copies of this vector B and two copies of this vector A. Well, the top here shows me A plus B equals a result, while the bottom shows me B plus A equals exactly the same result. This is the same thing that if I add 2 plus 3, it equals 5, but 3 plus 2, swiping those two things around, gives me exactly the same value. This also gives us a new method of looking at the vectors called the parallelogram method. If I have two vectors A and B, I form those into the size of a parallelogram, and the diagonal forms the result of adding those two vectors together. So we summarize the parallelogram method of graphical addition by recognizing that it's really the combination of using the tailed ahead for A plus B and B plus A at the same time. And since those two vectors add up to give us the same thing, no matter which order we place them in, the diagonal of our parallelogram is the sum of both A plus B and B plus A. Remember, graphical vector addition is just one method that we're going to use, and it gives us an overall picture of what it is that we're trying to do when we add two vectors.