# Question Video: Determining the Coordinates of Two Vertices of a Given Solid Shape Drawn in the Cartesian Coordinate System Mathematics

In the figure shown, the points π and π΄ have coordinates (0, 0, 0) and (7, 5, 6) respectively. Determine the coordinates of π΅ and πΆ.

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### Video Transcript

In the figure shown, the points π and π΄ have coordinates zero, zero, zero and seven, five, six respectively. Determine the coordinates of π΅ and πΆ.

We are told that the point π is the origin with coordinates zero, zero, zero. Letβs put that on our diagram, like so. And we can do the same thing for the point π΄ which has coordinates seven, five, six like so. As these points are in three-dimensional space, we need three coordinates to represent their locations. We therefore expect that π΅ and πΆ are going to have three coordinates too.

Before we try to work out the value of these coordinates, I think itβs helpful to go back and look at the two-dimensional case to see what these coordinates represent. Here is our representation of the more familiar two-dimensional space with two axes, just π₯ and π¦. And we have a point in this two-dimensional space, the point π with coordinates four, three. Now what do the values of those coordinates, four and three, tell us? First, itβs helpful to add in the origin π with coordinates zero, zero. And weβre going to see that the coordinates of the point π, four and three, tell us how to get to the point π from the origin π.

Starting at the origin, the first coordinate, the π₯-coordinate four, tells us that we need to walk four units in the π₯-direction. And the second coordinate, the π¦-coordinate three, tells us that we need to move three units in the π¦-direction; that is three units parallel to the π¦-axis. And having done this, weβve reached our point π. We could just have easily done it the other way around, first moving three units in the π¦-direction and then four units in the π₯-direction. It doesnβt matter which direction we go first.

Going back to our three-dimensional picture, letβs think about what the coordinates of π΄ tell us. Well, they tell us how to get to the point π΄ from the origin which I have marked. The first coordinate of the point π΄, the π₯-coordinate seven, tells us that we need to move seven units in the π₯-direction, in this case, along the π₯-axis. The second coordinate of the point π΄, the π¦-coordinate five, tells us that we need to move five units in the π¦-direction; that is five units parallel to the π¦-axis. The third coordinate is the new coordinate that we didnβt have in the two-dimensional case; it is the π§-coordinate. And the fact that the π§-coordinate is six tells us that we have to continue our journey by moving six units in the π§-direction; that is six units parallel to the π§-axis. So that is the interpretation of the coordinates of π΄.

I should just mention at this point that weβve been using the fact that the solid in our figure is a cuboid or right-rectangular prism. And therefore, all these sides are parallel to one of the axes. For example, π·π΅ is parallel to the π¦-axis and π΅π΄ is parallel to the π§-axis. So now that we have clarified what the coordinates of a point in 3D space mean, weβre ready to determine the coordinates of π΅ and πΆ.

First, letβs find the coordinates of π΅, which is the same as asking how to get to the point π΅ from the origin π. And we are going to need three coordinates: the first coordinate, the π₯-coordinate, tells us how far we have to go in the π₯-direction; the second, which is the π¦-coordinate, tells us how far we have to go in the π¦-direction; and the third, the π§-coordinate, tells us how far we have to go in the π§-direction. Well, if we look at our figure, we can see that we already have a path to π΅ from the origin; itβs through the point π·. From the origin π, we first travel seven units in the π₯-direction to the point π·. And once we are there, we just travel five units in the π¦-direction, and we get to the point π΅. So what are the coordinates of π΅? We travelled seven units in the π₯-direction, so the π₯-coordinate is seven and then five units in the π¦-direction, so the π¦-coordinate is five and then we were already at π΅, so we didnβt have to travel any distance in the π§-direction, so our π§-coordinate is zero.

And how about the point πΆ? We havenβt already drawn the path to πΆ from the origin, but we notice again that we can go through π·. And if you recall, we said that the length of the edge π΅π΄ was six. And as we are in a cuboid or right-rectangular prism, the length of π·πΆ must be six also. So how do you get to πΆ from the origin? First, you travel seven units in the π₯-direction, so the π₯-coordinate is seven. We donβt have to move at all in the π¦-direction, and so the π¦-coordinate is zero. We just have to move six units in the π§-direction, so the π§-coordinate is six. So thatβs our answer.

Weβve determined the coordinates of π΅ and πΆ. You might like to check that going in a different route from the origin π to the point π΅ or πΆ gives the same values of the coordinates as in the two-dimensional case.