In this question. We've got a shopper pushing a cart down a Siris of aisles, making two turns before the motion is complete and were asked to find what the shoppers A total displacement is, Um, and there is a no in the question that there could be more than one correct answer. And in fact, there's actually four correct answers. Four possibilities. Because if you think about it each time the shopper turns 90 degrees, there's two directions in which they can turn. So, for example, if you take a look at the first scenario here, the shopper walks 40 meters south, and then they can either turn 90 degrees to and go to the right or turn 90 degrees and go to the left. And then, of course, for the second turn, they can turn up or down. So combining those combinations for the turns, we get four different scenarios here. Okay, on What we need to find is the result in factor for each of these scenarios. So remember that the resultant vector will points from the starting point to the finishing point. So I'll just draw these on so you can see what we're trying to find here in all of these cases. And the good thing about these four scenarios is there is a little bit of overlap, right? So you can kind of see that the magnitude in A and B of for the displacement will be the same, and the manner to between C and D will be the same. And there's also a little bit of, um, a little bit of overlap with the directions as well, which we can discuss. So let's start by finding the magnitude. So we'll start with scenario and be and we'll find the magnitude of these two vectors, which, as already mentioned, should be the same now, in terms of the X direction, it's clear that the components of this result in factor is just the 15. That's the only vector that contributes to the X component here. And then the y component would be 40 minus 20. Okay, So, in order to calculate the magnitude of the displacement here, um, so this is for A and B. The magnitude will be the square roots of 15 squared, which is the X component plus 20 squared, which is the y component. Now, of course, this vector B points in the opposite direction for the X direction. But we don't need to worry about that in terms of the magnitude. So when we solve this on the calculator, we get that the displacement here is 25 everything is measured in meters. So the displacement, the magnitude of the displacement is 25 meters. Okay. And so what about the directions here? So in the diagram, these two angles here have to be the same gray. This is basically A and B are basically just a flipped a version of one another. If we flip along the 40 meter line, right, hopefully kind of do a mirror flip along the 40 degree line. We can translate A and two B and B into a. So the two angles that I marked in red here are the same. The only difference is you know, is it directed east or west? So we'll take into account that after we calculate this angle. So let's calculate the angle here. In order to do that, we will use tan okay, and we're using opposite over. Adjacent right, So opposite to this angle is 15 and adjacent to this angle is 20. And so when we take the tan inverse of that and plug it into the calculator, we look ants 36 degrees. So this is where the difference between the two comes in. What is it? 36 degrees with respect to write? Well, it's 36 degrees with respect to the South direction, and we swing either east or West. So for scenario A, the angle will be 36 degrees east of south. Another way to say that is that it is 54 degrees south of east. We can always just flip it around and do 90 minus, um, to get the other way of stating it. So basically, what we're saying in that case is that you know, we're measuring the angle with respect to east, and we had to swing, you know, 54 degrees down to get um, to get the angle. With respect to east and then for Scenario B, the angle can be stated as 36 degrees west of south, or it can also be stated as 36 0 sorry, not 36 but 54 degrees south of west. So the final answers here our displacement for a will be 25 to 25 meters, 54 degrees south of east. And for scenario B, we've got 25 meters, 54 degrees south of west. Okay, so those of our 1st 2 options and now we have to deal with C and D. So again they have the same magnitude, and angles actually are the same. They're just measured with with respect to different directions on the compass. So let's start with the magnitude. Okay? So the Y component of the displacement here will be 40 plus 20 this time, So that would be 60. Sorry, in the X component here is the 15 because that's the only rector that points in X direction. And so we plugged that into the calculator. We could see the magnitude here would be 61.8 meters. Okay. And we'll do the same thing to get our angle. So this time we're kind of looking at, you know, this big triangle. We're looking at the triangle that the displacement and the displacement in the why and the dissuasion in the X makes, and we're trying to find this angle here. Right? So the opposite side is 15 meters the adjacent side here is 60 meters, and so the angle is tan and verse of 15/60. Putting that in the calculator, we gets that the angle is 14 degrees. Okay, And again, we have just a slight difference in how that that angle is measured with respect to north, South, east and west. So for scenario. See, here it's 14 degrees, 14 degrees east of south and for Angle D here it's 14 degrees west of south, Uh, and we can also switch that around to measure it with respects to east and west instead of with respect to South. So if we want Teoh have a look at this angle instead, then we can just do 90 minus data. Okay, so all right, that outs. So we can you can see all of the different options ways of stating at here. So angle see is 14 degrees east of south yes, or 76 degrees south of east. And then, for Scenario D, we've got either 14 degrees west of south or another way of writing that is 76 degrees south of west. Okay, so in summary, displacement see is 60 1.8 meters 76 degrees south of East and Displacement D is 61.8 meters, 76 degrees south of west. So these four options here are the final answers. These are all of our options, all of our options for a displacement.