The points 𝑎, 𝑏, 𝑐, 𝑑, and 𝑒 are shown in the diagram. Now, this question has eight small parts to it. So let’s go through them one by one and answer each part as we go. Let’s start with the first one: how far north of 𝑎 is 𝑐?
So what we’re trying to do is to find out how far north of 𝑎 — this is 𝑎 — is this point 𝑐. Now, 𝑎 here is at the origin of the axis that we’ve drawn. So let’s imagining we’re travelling from 𝑎 to 𝑐. Obviously, we could travel directly there this way. But we can also break up our journey into components. We can break it up into the northbound component, which is this, and the eastbound component, which is this.
Now, for this question, we need to find out how far north of 𝑎 is 𝑐. In other words, the only distance we’re interested in is this northbound distance here. In other words, we can ignore the eastbound distance that we travel to get from 𝑎 to 𝑐. So we’re only worried about the northbound distance. And that distance happens to be this distance which is one meter as we’ve been told here plus this distance which is two meters which we’ve been told here. So the total northbound distance is one meter plus two meters which is three meters. And hence, how far north of 𝑎 is 𝑐? Well, 𝑐 is three meters north of 𝑎. And that’s the first part of our question answered.
Let’s look at the second part. How far west of 𝑎 is 𝑏? So here’s 𝑎 at the origin once again and here’s 𝑏. If we wanted to travel from 𝑎 to 𝑏, we could travel first westwards right this distance and then we could travel northwards. So first we go west and then we go north. But again for this part of the question, we don’t care about the northbound distance. We only care how far west we have to travel because we’re trying to find out how far west of 𝑎 is 𝑏. In other words, we want to find out this distance here. But we’ve been already told this distance. It’s five meters. Therefore, five meters is the distance that we’re looking for. In other words, 𝑏 is five meters west of 𝑎. And that’s the second part of our question answered.
So let’s look at the next two small questions. Firstly, how far south of 𝑎 is 𝑒? How far south of 𝑎 is 𝑒? Again, we can break this journey up into two components. Starting at the origin, we can go southbound because we’re starting at 𝑎 and trying to get to 𝑒 and then we can go eastbound. But once again, we only care how far south we go from 𝑎 to 𝑒. We don’t care about the eastbound component. And so we want to find out this distance here. And once again, we’ve been told that in the diagram this distance is four meters. Therefore, 𝑒 is four meters south of 𝑎.
Okay, so let’s move on to the next bit. How far east of 𝑎 is 𝑑? How far east of 𝑎 is 𝑑? Hmm, strange! 𝑑 is not east of 𝑎 at all. In fact, 𝑑 is west of 𝑎. We have to travel westwards first and then we have to travel southwards. And remember east is this direction — to the right of 𝑎 as we’ve drawn it in this diagram. But in order to get to 𝑑, we have to travel the other way. We have to travel west. But basically, travelling east or travelling west means you’re travelling along the same line. You’re just going in opposite directions. Therefore, going west is the same as going east in a negative direction.
Now, that might seem confusing at first. But think of your number lines. Let’s start with zero in the middle here. And if we want to go into the positive numbers and we want to get larger, then we travel towards the right, whereas if we start at zero and travel towards the left, we’re going to negative numbers. In other words, if travelling right is positive, then travelling left is negative. And again, we’re travelling along the same line just in the opposite direction.
So in other words, 𝑑 is this distance west of 𝑎. And that distance is already labelled in the diagram. It’s nine meters. So once again, 𝑑 is nine meters west of 𝑎. Therefore, 𝑑 is negative nine meters east of 𝑎. In other words, you need to travel negative nine meters east of 𝑎 to get to 𝑑. Obviously, you need to travel south as well. But in this question, we’re only looking at the east-west component.
Okay, anyway, let’s move onto the next part: how far north of 𝑒 is 𝑐? So this time, we want to find the distance north from 𝑒 to 𝑐. We’ll no longer starting at 𝑎, which is at the origin. However, this is not a problem. Once again, we can imagine that we’re going from 𝑒 to 𝑐. To go from 𝑒 to 𝑐, we can break our journey up into two components again. We start here and go northwards and then we need to go eastwards. So we want to find out the northwards component, which is this distance here.
And that distance happens to be the four meters that we’re told here plus the one meter that we’re told here plus the two meters that we’re told here. So that’s four meters plus one meter plus two meters. That total distance is seven meters. Now, of course, this orange arrow is meant to be one continuous thing. I just didn’t want to go over the axis. Anyway, so 𝑐 is seven meters north of 𝑒.
The next part asks us how far south of 𝑒 is 𝑑. How far south of 𝑒 is 𝑑? So starting at 𝑒, we want to go southwards first and then of course we want to go to westwards. But we don’t care about the westwards distance. Once again, we only care about how far south we have to go — so this distance. And this distance we’ve already been told is two meters. So 𝑑 is two meters south of 𝑒.
At this point, we’ve done six of the eight little questions. So let’s look at the final two.
Firstly, how far east of 𝑑 is 𝑏? How far east of 𝑑 is 𝑏? Now, this time we’re looking relative to 𝑑. So we start at 𝑑 and we go eastwards first until we go northwards. But yet again, we only care about the eastwards distance — this distance here. Now, this distance is a little bit more tricky to find out. We haven’t been explicitly given that distance anywhere in the diagram. However, we do know the distance from 𝑑 to the axis is nine meters. That’s what we’ve been told here and we know the distance from 𝑏 to the axis which is five meters, which we’ve been told here.
So the distance that we’re looking for which is this distance must be nine meters minus five meters. And that distance ends up being four meters. So this distance here from 𝑑 to 𝑏 eastwards is four meters. In other words, 𝑏 is four meters east of 𝑑.
Very finally, we’ve been asked how far west of 𝑏 is 𝑐. How far west of 𝑏 is 𝑐. Now, this scenario is something similar to what we saw earlier. 𝑐 is not west of 𝑏 at all. In fact, 𝑐 is east of 𝑏. Because if we wanted to go from 𝑏 to 𝑐, if we want to go from 𝑏 to 𝑐, we start at 𝑏 and then we travel eastwards first and then we go northwards until we get to 𝑐.
But we’ve been asked how far west of 𝑏 is 𝑐? So yet again, if we travel eastwards, then that’s the same as travelling west in the negative direction. In other words, we need to find out what this distance here is — this total distance. And because we’re travelling to the east, that’s the same as saying we’ve travelled negative that distance to the west.
So this total distance is equal to this distance here which we’ve been told is five meters plus this distance here which we’ve been told is seven meters. So this total distance is five meters plus seven meters which is 12 meters altogether.
However, remember we’re travelling east. So in other words, we’re travelling negative 12 meters to the west. And hence, 𝑐 is negative 12 meters west of 𝑏. And at this point, we’ve reached the end of our question.