Hey, guys. So many times when you're working on your vectors problems, you'll see that problems use different words to describe the directions of vectors. That's what we're going to check out in this video. We'll see that there are really just a few keywords to look out for like counterclockwise and clockwise and then northeast, southwest. We're going to check out those examples. So let's just get to it.
What you need to know about clockwise or counterclockwise angles, or sometimes abbreviated as CCW, is that these are positive angles. So, wherever axis you start from, as long as your angle is measured in this direction here against the clock, it's going to be positive. So, for example, if this is 45 degrees, it's going to be a positive 45 degrees. Now, clockwise angles, which are abbreviated by CW, are negative. So, for this b vector over here, this is a clockwise angle. It's negative. So, if this was 60 degrees, it's actually negative 60 degrees. Now, even though the positive or negative sign indicates just where it's going counterclockwise or clockwise, the reference angle that we use for our component equations, cosθ and sinθ, is always going to be a positive number relative to the nearest x-axis. Even though this is a negative 60 degrees, which all it does is tell us that it's a clockwise angle, the θx that we use is going to be positive 60 degrees.
Let's just go ahead and do some calculations here. We're going to draw each vector and calculate the components. So, we've got 5 meters at positive 37 degrees from the negative x-axis. So, it's important to pay attention to the signs of all of the information that's there. Positive 37 means we're going to be going counterclockwise like this from the negative x-axis. This is negative x, this is negative y. I'm going to start here and I'm going to go 37 degrees this way. So, this vector would be a equals 5 in this direction. So, if I wanted my x components and my y components, then I just have to use 5 times the cosine of 37 and I get 4. Then I have 5 times the sine of 37 and I get 3. We know we're in the 3rd quadrant, so these just pick up negative signs because these components point to the left and down.
Let's move on to b. Now we have a clockwise angle from the positive y-axis and it's 53 degrees. So we're going to have this vector here, b, 53 degrees clockwise from the positive y-axis. So this is my positive y, positive x. And so this angle is going to look like this. So we're going to look at 53 degrees and this is going to be a negative sign over here. So this is my b vector which equals 5. Now, we just have to calculate the legs or the components over here. My By and my Bx. We know that Bx is just going to be B times the cosine of theta. The problem is, is that I have this negative angle here that's relative to the y-axis. I can't use that. I have to figure out what the complementary angle is, which is 37 degrees. So I'm still just going to use 5 times the cosine of 37 because that's the reference angle. It's positive relative to the nearest x-axis and that's just 4, and it's positive because it points to the right. And then my By is 5 times the sine of 37, which gives me 3. It's also positive because I'm in the 1st quadrant. Alright.
The other thing that you might see is you might also see cardinal or compass directions like northeast, south, and west. So you're going to see directions described like 30 degrees north of east. And this is actually really straightforward. You're basically going to go from here, you're going to curve towards this. So we're going to draw each vector and calculate just the x component for these two examples. Let's just get to it. So a equals 6 at 30 degrees North of East. So we're going to start basically by drawing the arrow in the second direction. Now you're just going to curve towards the first, which is to the North. So this is A equals 6 and this results in an x component of 6 times the cosine of 30, which ends up being actually 5.2.
What about b? Now, we have 10 at 53 Degrees West of South. So again, this is our second direction. We're going to draw the arrow in the direction of the first one, which is south, and then we're going to curve 53 degrees towards the west. This is 53 degrees like that and this is our arrow. We know b equals 10. So I want to calculate the x component over here. I have to use b times the cosine of 37 degrees (our reference angle). That's my reference angle, which we figured out. We're going to get 8, and remember to include a negative sign because west is conventionally chosen to be negative.
Alright, guys. That's it for this one. Let me know if you have any questions.