Ruth sets out to visit her friend Ward, who lives 50 mi north and 100 mi east of her. She starts by driving east, but after 30 mi she comes to a detour that takes her 15 mi south before going east again. She then drives east for 8 mi and runs out of gas, so Ward flies there in his small plane to get her. What is Ward's displacement vector? Give your answer (a) in component form, using a coordinate system in which the y-axis points north, and (b) as a magnitude and direction.

Video transcript

Hey, everyone in this problem, we're told that Jack and Joy are business partners. Hey, Joy resides 80 kilometers south and 60 m west of Jack's residence. OK. So we're gonna come back to the rest of the question in just a minute. But let's start by drawing at what we have. OK. So if we think about our directions, OK. We're gonna have north pointing upwards east, pointing to the right south, pointing downwards and west pointing to the left. Ok. So we're gonna start at Jack's Resonance and join lives 80 kilometers south. OK. So we're gonna go downwards 80 kilometers, it's 60 m west. We're gonna go to the left 60 m and this is going to be Joy's residence. All right now, getting back to the question, it says that Jack drives 42 kilometers south. Ok. So let's draw this out. So Jack is going to drive 42 kilometers. So, so downwards 42 kilometers then 12 kilometers east and so to the right 12 kilometers. Ok. We're not quite to scale here, but we're gonna label everything. So we know our distances and then another 22 kilometers south. Ok? So downwards again, 22 kilometers until he reaches a meeting point. Ok. So this is going to be the meeting point. Now, we're told that Joy is gonna use a helicopter to arrive at the meeting point. Ok. So Joy is going to go directly to that meeting point and we're asked to determine her displacement vector. OK. So this vector we've drawn in blue going directly from Joy's residence to the meeting point which we'll call d that's the displacement vector that we are looking for. Now, we're told to express our answer two ways first using components and were the X axis points east and the positive Y axis points north and then using magnitude and direction. OK. So we're given four answer choices A through D each containing a different combination of answers for part one and part two. All right. So we have this big messy diagram that we've drawn, but we wanna simplify, OK. What we're interested in is Joy's displacement vector that we've drawn in blue. So let's go ahead and just draw that portion of our diagram. So we have her residence and we have the meeting point and the displacement vector is going to be an arrow pointing between the two. Now let's break this up into the X and Y components. OK. Now, if we think about the X distance, OK, going from Joy's residence to the meeting point, what do we have? Well, we have the 60 m from Joy two, the line where Jack's resonance is, and then we have the 12 kilometers that jack traveled east. Ok. So the X direction here is going to be that 60 m plus the 12 kilometers. If we add that up at 60 m, we can convert into kilometers by dividing by 1000. And so this is gonna be 12.06 kilometers. Now we're gonna do a similar thing for the Y direction. Now, in the Y direction, what do we have? Well, the distance from the vertical distance from Jack to Joy is 80 kilometers. Ok. The meeting point jack traveled 42 kilometers downwards and then another 22 kilometers downwards to get to the meeting point. OK. So this distance, this vertical distance that's left over is gonna be that total distance of 80 kilometers minus the 42 kilometers jack traveled minus the additional 22 kilometers jack traveled. OK? And when we work this out, we get that this is going to be 16 kilometers. OK? So again, our triangle is not to scale this vertical distance is actually larger. So let's adjust that. Um So that it's clear that that's the case or at least closer to the case. All right. So now we have the X and Y component of this displacement vector. So we can answer part one of this question. OK. So for part one of this question, it was asking us to write the displacement vector in terms of the X and Y components. Now we know that the X component is 12.06 kilometers. It's pointing to the right. OK. So in the positive X direction, so we have positive 12.06 I have in the Y direction, we have 16 kilometers. It's pointing upwards. That's our positive Y direction. So we have positive 16 J hat. You remember I hat corresponds to the horizontal or X direction J hat corresponds to the vertical or Y direction. And these are both in kilometers. All right. So that's our answer for part one, we just needed to figure out what those X and Y components are. We take a look at our answer choices. Option A and option D have this displacement vector. Option B and C both have that the X component is 72. That's not what we found. OK. So B and C we can eliminate and we're down to two possible answer choices. Now moving to part two, we wanna write this in terms of the magnitude and direction. OK. So let's think about the magnitude and we're gonna call the magnitude of our displacement vector D. OK. And we can see that we have a right angle triangle. So to find D we can use the Pythagorean theorem since we know the other two sides. OK. So let's go ahead and do that. So Pythagorean theorem tells us that D squared is going to be equal to the sum of the squares of the other two sides. So 12.06 kilometers squared plus 16 kilometers squared. And we can simplify on the right hand side. And we have that D squared is 401 0.4436 kilometers squared. And finally taking the square root to get that, the magnitude of this displacement vector D is going to be 20.036 kilometers. OK? And again, that is our magnitude. All right. So that's part of part two. OK. Let me just put that we're doing part two. Now. All right. The second part of part two is to find the direction. So what we wanna find is the angle theta. OK. So the angle theta is gonna be between our displacement vector and that horizontal. OK. Now we know the opposite and adjacent sides. So we're gonna use a tangent to relate the two. So recall that tangent of theta is equal to the opposite side divided by the adjacent side. OK. So substituting in our values, the tangent of the is going to be equal to 16 kilometers divided by 12.06 kilometers. We can take the inverse tangent of both sides and we get that the, it's going to be about 52.99 degrees. OK. So this is going to be our direction since it is already measured from that positive x axis. OK. All right. So that's all we needed for part two, we have our magnitude, we have our direction. So let's take a look at our answer. Choices. We had narrowed it down to option A or option D option A has that. The magnitude is 20 kilometers and the direction is 53 degrees north of east. That is exactly what we just found. And so option A is going to be the correct answer. Ok. Option D had the incorrect magnitude and direction. Thanks everyone for watching. I hope this video helped see you in the next one.