Let θ be the angle that the vector A makes with the +x-axis, measured counterclockwise from that axis. Find angle θ for a vector that has these components: (a) Ax = 2.00m, Ay = −1.00 m;

Video transcript

Hey, everyone in this problem, we're asked to use a convention where theta is a counterclockwise angle measured from the positive X axis to a vector in order to determine the angle for vectors of the following components. OK. So in part A, we have MX is equal to 3 m and my is equal to negative 1 m. And in part B, we have NX is equal to negative 5 m and NY is equal to negative 3 m. We have four answer choices A through D, each of them containing a different combination of angles for M and N. And we're gonna come back to those once we're done working through this problem. So let's start with part A and in per A, we're told that the X component is going to be 3 m. The Y component is going to be negative 1 m. If we draw axis, there are X and Y axes, we have that the X component is gonna go out 3 m, the component is negative one. So it's gonna go down 1 m. And so our vector M is going to point from the origin to the, this point at three common negative one. OK. This is our vector M. All right. Now we want to take the as a counterclockwise angle measured from the positive XX. OK. So what we want is the measured from the positive X axis counterclockwise all the way around to add. OK. So in green, I've drawn what data is going to be, what's an easier angle to calculate is going to be alpha. OK. So I'll draw alpha and bullet and also is gonna be a little bit easier because it is a an acute angle and it's related to this 3 m, this 1 m, those sides of our triangle that we have here, we're gonna be able to calculate alpha and use that to find the. Now you can see from the diagram that theta and alpha together make up one complete rotation. And so we know that theta was alpha is going to be equal to 360 degrees. There are 360 degrees in a circle. OK? So if we can find alpha, we'll be able to calculate beta later, it is going to be equal to 360 degrees minus. All right. So let's think about alpha here. We know the opposite side and we know the adjacent side. OK? Based on the components of the vectors we were given. So let's use the tangent and that's gonna relate those two sides that we have. So we have that tangent of alpha is going to be equal to the opposite side divided by the adjacent side, which is gonna be equal to one third. And we're using just the positive values here, we're just talking about the magnitude of these sides because we're gonna take that acute angle between zero and 90 degrees. All right. So we can calculate alphabet by using the inverse tangent 01 3rd when we get that alpha is equal to about 18.43 degrees. So now if we go back to calculating beta, which is what we actually want to calculate like beta is gonna be equal to 360 degrees minus 18.43 degrees. So we get that data is equal to 341.57 degrees. All right. All right. So we're done with part A, let's move to part B and in part B we're gonna do the exact same thing. We just have different components to work. OK. So in part B, let me just write the component. So we can see them, we have the NX is going to be negative 5 m and NY is going to be equal to negative 3 m. All right. So we're gonna draw our axes X and Y and now we're gonna draw our vector N and the X component is going to be negative five. The Y component is going to be negative three. We put a point at that coordinate negative five comma negative three and then we draw an arrow from the origin to that point. And that is our vector N. In this case, the again is gonna be measured from the positive x axis. It's gonna go all the way around to our vector. So in green, we've drawn the up again in alpha. Again, we want alpha just to be an acute angle. This is our reference angle. And so that's gonna be the angle between the negative x axis and our vector end. So in this case, the and alpha are related in a different way than they were before, we know that if we go all the way from the positive X axis to the negative X axis, that's 180 degrees. In this case, we're going 180 degrees and then alpha degrees more. So in this case, data minus alpha is going to be equal to 180 degrees. OK? And drawing a diagram of these vectors really helps you try to figure out how these angles are related. OK? Because it's not always the same like we're just seeing in part A and part B here. So if you have a diagram, you can kind of visualize how those angles are related. OK. So we wanna calculate data. So we're gonna calculate data 180 degrees plus alpha. OK. So again, we're gonna find alpha first reference angle and then we're gonna come back to the now our reference angle again, we're gonna be using tangent tangent of alpha is gonna be the opposite side divided by the adjacent side. And in this case, the opposite side is three, the adjacent side is again using positives because we're just looking at the magnitude here that acute angle. All right. So that tells us that alpha can be calculated through the inverse tangent of 3/5 which gives us an alpha value of about 30.96 degrees. OK. Now we have alpha, we go back to theta. Don't forget what the question is asking for. It can be really easy when you find this angle to stop and think that that's your answer. But remember what the question is asking and we need the angle theta instead of alpha, the theta is gonna be 180 degrees plus this angle alpha that we found 30.96 degrees, which gives us a data value of about 210.96 degrees. All right. So let's go America and we're gonna take a look at our answer choices and all of our answer choices have one decimal place. So we're going round to one decimal place. And what we can see is that the correct answer is going to be option C and we have M is equal to 341.6 degrees and N is equal to 211 degrees. Thanks everyone for watching. I hope this video helped see you in the next one.