A postal employee drives a delivery truck over the route shown in Fig. E1.25. Use the method of components to determine the magnitude and direction of her resultant displacement. In a vector-addition diagram (roughly to scale), show that the resultant displacement found from your diagram is in qualitative agreement with the result you obtained by using the method of components.

Video transcript

Welcome back everybody. So we have Aaron taking this route from his house to the library and we are asked to find the magnitude of the resultant displacement and its direction. Well that vector is going to be the direct from the beginning to the end. So this is our displacement factor. I'm going to call it R. And just for reference I'm gonna call all these tiny vectors, A, B and C. Well, in order to find the magnitude and direction of a vector, it's given by these formulas right here. So the magnitude of a given vector let's say A is going to be the square root of its X component squared plus its Y component squared in the direction that we are looking for is going to be equal to the arc tangent of its Y component divided by its X component. But what are the Y and X components of our Well it's as easy as this you're gonna have that the X component of our is just going to be the X component of a plus the ex opponent of B plus the X component of C. And the why component of our is going to be the same thing except with all of the Y components. So let's go ahead and find our X and our Y. Keeping in mind that we have to find the magnitude and direction of our our Alright so our X let's start there. So our X is equal to the X component of a. Well here's our a vector and this is a little bit hard to see. So I'm gonna bring this out. The angle that it makes with the ground is 30 degrees and we are looking for its x component. So this is going to be equal to the magnitude of the vector 1.7 times the cosine of the angle 30. Right? So then plus the X component of B will be is only acting in the X direction. So that's just gonna be the magnitude of the vector. Plus the X component of C. C is only acting in the Y direction. So there is no X component of C. Let's go ahead and find Ry now so are Y is equal to a Y. Well this time we're looking for the y component of a so it's going to be equal to the magnitude times the sine of plus the Y component of B will be, is not acting in the Y direction. So there is gonna be no Y component or plus the Y component of C. C is acting solely in the Y direction. So it's just going to be the magnitude of that vector. When you plug these into your calculator, you get 7.27 for the X component and 2.25 for the Y component. So now we are ready to use these formulas to find the both magnitude and direction of the displacement. Let's start with the magnet here, we are going to have that the magnitude of r is equal to the square root. Its X component 7.27 squared plus its y component of 2.25 squared. This is going to be equal to when you plug into your calculator 7.61 kilometers. Now its direction is given by the arc tangent of its Y component 2.25 divided by its X. Component. And when you plug this into your calculator, you get 17.20 degrees, giving us the magnitude and direction of our displacement vector corresponding to answer choice. D. Thank you guys so much for watching. Hope this video helped. We will see you all in the next one.