A plumber steps out of his truck, walks 55 m east and 38 m sou...

Question

A plumber steps out of his truck, walks 55 m east and 38 m south, and then takes an elevator 12 m into the subbasement of a building where a bad leak is occurring. What is the displacement of the plumber relative to his truck? Give your answer in components. Also give the magnitude and angles, with respect to the 𝓍 axis, in the vertical and horizontal planes. Assume 𝓍 is east, y is north, and 𝒵 is up.

Accepted Answer

Welcome back. Everyone in this problem from a base camp. A mountain climber begins her ascent moving 800 m toward the north and then 600 m toward the east to reach an intermediate camp. The next day, she climbs 400 m vertically to reach the summit. Expressing your answer in components determine displacement of the climber from the base camp. Additionally, in both the vertical and horizontal planes get the magnitude and angles with respect to the X axis. We are considering X is east, Y is north and Z is up for our answer choices. A says the displacement components are six is 600 I plus 800 J meters. The magnitude is one multiplied by 10 cubed meters. The angle in the horizontal plane is 53.1 degrees and the angle in the vertical plane is zero degrees. B says the displacement components is 600 I plus 800 J plus 400 K meters. The magnitude is one multiplied by 10 cubed meters. The angle in the horizontal plane is 45 degrees and the angle in the vertical plane is 30 degrees C says the displacement components is 400 I plus 400 kilometers. Its magnitude is 1.45 multiplied by 10 cubed meters. The angle in the horizontal plane is zero degrees and the angle in the vertical plane is 15.8 degrees. And finally, D says the displacement components are 600 I plus 800 J plus 400 K meters. The magnitude is 1.08 multiplied by 10 cubed meters. The angle in the horizontal plane is 53.1 degrees and the angle in the vertical plane is 21.8 degrees. Now, first, let's try to make sense of what's really, what's really happening here. OK. So we know that our mountain climber begins her ascent and we're talking about an ascent in the three dimensional plane. OK? And the movement towards the north and the east followed by the vertical climb can be represented as vector components in a three dimensional coordinate system. So let's just sketch quickly a 3D system here. OK. Right. And in our 3d system, we have our axis I for X axis J for our Y axis and K for our Z axis. OK. Now, what do we know? Well, starting from our base camp, OK. The mountain climber moves 800 m toward the north and we're considering Y to be the north. So that means towards the north, we'd move 800 m on the Y axis. Next, it says that she moves 600 m toward the east. And we're considering XX to be east. So we're going to move 600 m east, probably ending up somewhere right here. After we get to that point, the next day, she climbs 400 m vertically to reach the summit. So that's going to be if I can, if you can picture your mind's eye on a 3d grid here, that's going to be 400 m on the Z axis. So now the displacement vector that represents her journey would be from our origin. OK? To this point that I've drawn and again, you can, I would suggest being creative trying to bear in mind that this is a 3d axis. I know it's on two D paper, but just to get the idea of what's going on here. So since she traveled 800 J, OK, that's 800 J, 600 I and 400 K, then the displacement vector D OK is going to be equal to 600 I plus 800 J plus 400 K meters. Those are the components of our vector. So now we've done the first part, we have the displacement of the climber from the base camp. Let me draw that, draw some red over it to show that, hey, this is, this is pretty important. Now, next, we're supposed to figure out the magnitude and angles we're supposed to get the magnitude of our displacement vector and its angles with respect to the X axis. In the vertical and horizontal planes. So let's see if we can figure that out. First, let's try to get the magnitude of our vector. Now, we have the components of our displacement vector and recall that to find the magnitude of our vector, we can find the square root of the sum of the squares of all its components. In this case, we're going to be finding the square root of X squared plus Y squared plus Z squared using that idea here, OK. That means that the magnitude of our vector is going to be equal to the square root, the square root a 600 m squared plus 800 m squared plus 400 m squared. And the terms are being squared, it's not the, the unit is going to be square meters. But we're saying it's going to be 600 m multiplied by 600 m and so on and so on. Now when we wear code or expression here, OK, we end up with a sum of, let me see here. This is going to on our calculator. You should be seeing it, but it's gonna be 1,160,000 square meters. And when we find the square root of our value here, then we get the magnitude of our displacement to be equal to 1077.03 m. If we write that in standard form, like it was written in the answers, we'll end up with 1.08 multiplied by 10 cubed meters. OK. So that will be the magnitude of our displacement since we've found the magnitude and we've found the displacement of the vector. All we need to do now is to figure out the angle in the horizontal plane and in the vertical plane with respect to the x axis. Let's start with the angle in the horizontal plane. Now here with respect to the X axis. OK. So with, let me put that in blue, I have to, I have to be a bit creative with space here with respect two the X axis. OK. Then the angle in the horizontal plane can be found with a tangent function. So the angle in the horizontal plane tan theta, OK? And let's say that that angle theta, the angling in the horizontal plane tan theta is going to be equal to the ratio of the Y component to the X component. Therefore, the anger is going to be the inverse tangent of Y divided by X. And here we know that Y is our J component which is 800 X is our I component 600. So our anger is going to be the inverse tangent of 800 divided by 400. If that's the case, then when we put this into our calculator, we'll be finding the inverse tangent or sorry, 4 800 divided by 600. My apologies. So you're gonna be finding the inverse tangent of 800 divided by 600 which is eight divided by six or four, divided by three. And when we do that, we should get an angle of 53.1 degrees to the nearest 10th of a degree. So that would be the anger between the horizontal plane and the angle theta in the horizontal plane for a displacement vector with respect to the x axis. Now, how about the Anger in the vertical plane with respect to the X axis? Well again as in blue here, the Anger in the vertical plane, OK. Angle in the vertical plane can also be found with the tangent function because considering the vertical climb the anger, let's call that anger. Uh let's call that angle Phi. OK. Then we can also find Phi with respect to the X axis in the vertical plane which involves the Z component. OK. Oops. So using the tangent, then the tangent of that angle tangent of Phi is going to be equal to the Z component divided by the square root of the sum of the squares of the X and Y component. So that means then that our anger phi is going to be equal to the inverse tangent of that expression where we know our Z component is 400 m and our X and Y components are 600 m. So that 600 m squared plus 800 m squared, that's our Y component. OK. And we're going to find the square root of that. No, we could work this through and through or we could also work this out using or uh put this expression directly in our calculator to get an exact value. OK. Now, when we do that, this expression works out to be equal to 0.4. OK. So we're really finding the inverse tangent of 0.4 and the inverse tangent of 0.4 is approximately equal to 21.8 degrees to the nearest 10th of a degree. Therefore, the anger that the displacement vector makes with the vertical plane with respect to the X axis is 21.8 degrees. Now, when we look at all of our answers, the displacement components are 600 I plus 800 J plus 400 K meters. Its magnitude is 1.08 multiplied by 10 cubed meters. The angle in the horizontal plane with respect to the X axis is 53.1 degrees. And in the vertical plane with respect to the X axis is 21.8 degrees. If we look on our answer choices, the correct choice is answer D. Thanks a lot for watching everyone. I hope this video helped.

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