Hey, guys. So previously, we saw that vectors are just triangles, and vector math turns into a bunch of triangle math. Well, you're going to need to know how to add vectors a lot in physics. So in this video, I'm going to show you how to add vectors graphically. I feel like it's a great way to visualize what's actually happening when you are combining multiple vectors. Let's check it out. So remember that vectors are drawn as arrows, like a displacement vector or something like that. And the way that we add vectors is we just connect those arrows, and we're going to do this in a way that your textbooks and professors call tip to tail. We actually saw this when we added perpendicular vectors. Let's check it out.

If you were to walk 3 meters to the right and then 4 meters up, all we did was we basically just connected these arrows tip to tail, and your total displacement was actually as if I had actually just walked from here to here. We called this c, and the way that we calculated this was we found that these arrows just made a triangle. We could use the Pythagorean theorem. So this magnitude here was the square root of 3 squared plus 4 squared, and that was 5 meters. So this total displacement that we found, which is just the shortest path, gets a fancy name. It's called the resultant vector sometimes or the resultant displacement. Sometimes you'll see c or r for that resultant, and basically, the resultant is always going to be the shortest path from the start of the first to the end of the last, just like we did over here. It's as if you had walked in this direction.

So one thing that we haven't seen yet is how to add vectors that aren't perpendicular. And so we've got these 2 vectors here, and to add them, we just have to use the tip to tail method. We just have to add or connect these things tip to tail. But notice that they both start at the same place. So one thing I can do here is I can basically pretend as if I can pull this vector, if I pull this thing over to the right like this so that I can connect them, Eventually, I want to line up so that I can connect these vectors tip to tail. And what you would get is something that looks like this. We know that vector a is going to be 2 to the right and one up. So that means my vector a is going to be 2 to the right and one up. It looks like this. And then my b vector is going to be 1 to the right and 3 up. So from the endpoint of here, I'm going to go 1 to the right and 3 up, and so I'm going to end up over here. So these are the vectors connected tip to tail.

And so the resultant vector, which is the total displacement, is the shortest path from start to finish. So here's the start of the first one and the end of the last one, and so this is my displacement vector here. So this is my resultant. And so the way I calculate this, the magnitude is I just break it up into a triangle, which I know the legs of this triangle are going to be these legs right over here. And I can get the numbers just by adding up the boxes and counting up the boxes. So I've got 3 here and 4 here. So that means that the magnitude is 32 + 42, which is equal to 5 meters.

So this is how I add together a and b. What if instead I want to add b plus a? So basically, I'm just going to do the opposite. I'm going to start off with the b vector first, which I know is 1 to the right and 3 up. So that means it's going to go like this. So this is my b vector. And then my a vector is 2 to the right and 1 up. So that means from here, I'm going to go 2 to the right and 1 up. So connect them tip to tail, and the displacement or the resultant is going to be the shortest path from the start to the end. So that means this is my displacement over here. It's c. I break it up into a triangle and get the same exact legs that I did before. I've got 3 and 4, and so the Pythagorean theorem, the 32 + 42, square rooted, is going to be 5 meters.

So, notice here that it didn't matter the way that we added the vectors together. A or a plus b or b plus a. We ended up at the same exact point. We ended up with the same exact arrow. So that means that adding vectors does not depend on the order. This is something that your textbooks call the cumulative property. It just means that 3+2 is 5, 2+3 is also 5. So that just means that it doesn't matter the way that you add vectors up together. You're always just going to get the same number.

Alright, guys. That's it for this one. Let's go ahead and get another example. We're going to find the magnitude of this resultant vector, a plus b. So we're just going to basically combine these vectors so that they are tip to tail because they're not right now, and then we just find out the shortest path, between start and finish. So I'm going to add a plus b. So I'm going to start off with a. And then b here is going to be 1 to the left and 4 down. So that means that from the end of b, I'm going to go 1 to the left and 4 down. So 1, 2, 3, 4. So that means that my b vector looks like this. So kind, I kind of just imagine that I've just pulled this b vector about this way to the right there so that they line up tip to tail. So I can kind of just pretend that this isn't there anymore. So now what's the resultant vector? Well, the start of the first one was here and the end of the last one is over here. So that means that my displacement vector is the shortest path and that's the arrow right here. So this is my displacement vector. We can break it up into a triangle. Basically, these are going to be the legs of this triangle, and I can count up the boxes. I've got 3 here and 3 here. So that means the magnitude of the c vector, which is usually written by a b