 So now we're going to take a look at momentum as a vector. Now that momentum equation was linear momentum was p equals mv, or the mass times the velocity is going to give us our momentum. In that equation, the mass is a scalar quantity, but velocity is really a vector. And when I multiply a scalar times a vector, I get a vector quantity, which means momentum should be a vector. That means if I want to write this out in full form, I need to specify that both momentum and velocity are vectors. Whenever we have a vector, we can break it down into the components for each direction. Now in this case, we have the possibility of having an x, a y, and possibly a z for our directions for the velocity. And if you have x, y, and z velocity components, you have the corresponding x, y, z momentum components. If I was going to write this out as a vector equation, again, our vector equations use the i hat, j hat, and k hat to help show us the direction. And so we just have the individual components in each one of those directions. I could also write it out as the mass times the velocity for each one of the components. If all I care about is the magnitude, then I could calculate that from my components. And this is going to be using my standard Pythagorean theorem. Again, if I only have x and y, I can leave off the z. That's my more familiar form of the Pythagorean theorem for a triangle. But if I do have three components, I square each of the components, add them together, and take the square root. But I can also relate this back to speed. Explicitly, the mass times the magnitude of the velocity, which is also known as the speed, will give me the magnitude of the momentum. So the magnitude of the momentum is the mass times the speed. And this is if I don't care about the direction. And this is actually where our original equation that we started off this presentation using. Because most textbooks don't use the absolute value signs to indicate magnitude, they simply use a plain font symbol without the vector area on top. Now if I do care about the direction, what's important to realize here is that whatever direction I have for the velocity, that's the direction I'm going to have for the momentum as well, because the mass will not affect the direction. What about change in momentum? Some of the equations we're going to be working with are going to use this delta p or the change in momentum. In that case, I need to keep in mind that this is a vector subtraction. So direction matters when I look at my final momentum compared to my initial momentum. In that case, it's probably best to work with the components or the equations, in which case I can find the change in momentum in the x direction separately from the change in momentum of the y direction. So that's how you treat momentum as a vector.