Hey, guys. We're going to be working with vectors a lot in physics, so you're going to have to get very good at how we manipulate and combine them. In this video, I want to introduce to you what vector math is and what it's all about. But I want to take a minute to describe what the next few videos are going to be about. We're going to be working with vectors, with these diagrams, with these little axes and grids, and boxes because it's a great way to understand very visually what's going on with vectors. Then later on, what we're going to see is we're going to see all of the math equations that describe vectors. So let's get to it.

Adding and subtracting scalars is pretty easy. Remember, scalars are just simple numbers. But vectors have directions, so math is sometimes not as straightforward. For example, if you were to combine scalars, let's say you're moving and you're combining a 3-kilogram and 4-kilogram box, then the way that you add these boxes together if you were to lump them together in a single box is you just add 3 and 4 straight up. So 3+4 is just 7. Combining scalars is just simple addition.

Where things get a little bit more tricky is when you start combining vectors, and there are really just 2 cases that you're going to see. You combine parallel vectors, then there would be like walking 3 meters to the right and 4 meters to the right. So, because vectors have direction, they're drawn as arrows. This 3 meters to the right and 4 meters to the right would look like this. They would just add together because they are both in the same direction, resulting in a total displacement of 7 meters.

Where things get a little trickier is when you're combining perpendicular vectors. So now let's say I walk 3 meters to the right and then 4 meters up. So we got this grid that's going to help us visualize what's going on here. Now, you might think that the total displacement is just 3+4, which is 7, but it's actually not because your total displacement is really just the shortest path from where you start to where you end. What happens is we make a little triangle like this. The way we solve for this is by using an old idea, an old math equation from algebra or trigonometry called the Pythagorean theorem. So this is the Pythagorean theorem: c 2 = a 2 + b 2 . This means my total displacement is 3 2 + 4 2 , which is actually equal to 5 meters.

We're going to continue with more examples. For instance, if we walk 10 meters to the right and then 6 meters to the left. The total displacement, after drawing the displacement vectors, is just 10 minus 6, which is 4 meters, because these displacements are antiparallel but along the same line. Now, for part B, if I walk 6 meters to the right and then 8 meters down, we have perpendicular vectors again, forming another triangle. The displacement, or the hypotenuse of this triangle, is solved using the Pythagorean theorem: c 2 = 6 2 + 8 2 , resulting in a displacement of 10 meters.

Alright, guys. That's it for this one. Let me know if you have any questions.