Hey everyone, and welcome back. So up to this point, we've spent a lot of time talking about vectors, and you may recall one of the first things that we learned about vectors is that they have a direction. Now in this video, we're finally going to be learning about how we can calculate the direction of a vector. This might sound a bit complicated or a bit scary, but don't worry about it. Because it turns out that calculating the direction of a vector is very similar to finding the missing angle of a right triangle, which is something we've already learned about. And this is a skill that is very important to have in this course, so let's just go ahead and get right into things. Let's say we have this vector here which we'll call vector v. Now, recall that the direction of a vector is the angle that the vector makes with the x-axis, so we need to figure out what this angle is right here.

If we had the x component as well as the y component of our vector, could you think of a clever way that we could figure out what this angle is? You need to recall a memory tool that we learned for right triangles, which is SOHCAHTOA. SOHCAHTOA teaches us that tangent is opposite over adjacent. So if we had the opposite and adjacent side of a right triangle, we could calculate the angle using this equation right here. This might seem a little bit out of the blue but notice something, the vectors we have here form a right triangle. So we can use this right triangle to figure out what this angle is. If the tangent of our angle is opposite over adjacent, that means that the tangent of our angle here, which would be the direction of our vector, would simply be the y component divided by the x component. If I wanted to find our angle theta, which would be the direction of this vector, I just need to take the inverse tangent of this fraction right here. So we have the inverse tangent of the y component divided by the x component. I can see here that the y component is 1, 2, 3 units up, and I can see that the x component is 1, 2, 3, 4 units to the right. So we'll have the inverse tangent of 3 divided by 4, which on a calculator comes out to approximately 37 degrees. This is not the exact result, but it is a very close approximation to the answer so this would be the direction of our vector. As you can see, it's pretty straightforward if we recall this memory tool for right triangles.

But it turns out there are going to be some problems you come across that don't give us as direct an answer where we can just plug it into our calculator. And to make sure that we know how to solve these examples, well, let's try them. Here we're told, drop the vector and find the direction of each vector expressed as a positive number from the x-axis. So let's go ahead and start with example a, where we have the vector 2 negative one. What I first need to do is draw this vector. Well, on the x-axis, this would be here at 2, and on the y-axis negative one is down there. So our vector is going to look something like this. Now, what I need to do is figure out the direction of this vector and I'm going to first calculate this angle right in here. Now recall to find this angle, we need to take the inverse tangent of the y component divided by the x component. So we're going to have the y component, which is negative one, divided by the x component, which is 2. And if you type the inverse tangent of negative one over 2, you should get approximately negative 27 degrees. So this right here is the missing angle. Now you might think that this would be the solution to the problem, but it turns out it's actually not. Because we need to express our answer as a positive number from the positive x-axis. So how do we find this angle? Well, we need to start on our positive x-axis and go counterclockwise all the way around until we reach this vector. I can see that this is close to a full 360-degree rotation. So our angle is going to be a full 360-degree rotation minus this 27-degree angle that we just calculated. So I have 360 minus 27 degrees which comes out to 333 degrees. So this right here would be the direction of our vector and the solution to this problem.

To really make sure we understand how to do these situations where we don't have our vector just somewhere here in the first quadrant, let's go ahead and try another example. In example b, we're asked to find the vector negative three negative three to sketch it and to figure out its direction. What I'm first going to do is draw this vector. You can see on the x-axis negative 3 is right there, on the y-axis negative 3 is down here, so our vector is going to look something like this. What I'm going to do is calculate this angle to find the direction and to find this angle, well, this angle is going to be the inverse tangent of the y component divided by the x component. θ = tan - 1 - 3 3 negative 3 divided by negative 3 is just going to give you positive 1. So we have the inverse tangent of positive 1 which is equal to 45 degrees. So this right here is this angle, α and if we want to find the total direction of our vector, we need to start on the positive x-axis and go counterclockwise until we reach our vector. Well, what would this be? Well, that's a 90-degree angle. That would be a 180 degrees total. And so we take 180 degrees and add it to this angle we calculated, that would be the total direction. So we have 180 degrees plus this angle which is 45 degrees. So a 180 plus 45 should give you an angle of 225 degrees. So this right here is the total direction of our vector and the solution to this problem. So that is how you can find the direction of vectors even if you are given a vector that is not in the first quadrant like we had in this first example. So this is the strategy. Hope you found this video helpful. Thanks for watching.