In the figure shown, the points 𝑂 and 𝐴 have coordinates zero, zero, zero and seven, five, six respectively. Determine the coordinates of 𝐵 and 𝐶.
We are told that the point 𝑂 is the origin with coordinates zero, zero, zero. Let’s put that on our diagram, like so. And we can do the same thing for the point 𝐴 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 𝐵 and 𝐶 are going to have three coordinates too.
Before we try to work out the value of these coordinates, I think it’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 𝑥 and 𝑦. And we have a point in this two-dimensional space, the point 𝑃 with coordinates four, three. Now what do the values of those coordinates, four and three, tell us? First, it’s helpful to add in the origin 𝑂 with coordinates zero, zero. And we’re going to see that the coordinates of the point 𝑃, four and three, tell us how to get to the point 𝑃 from the origin 𝑂.
Starting at the origin, the first coordinate, the 𝑥-coordinate four, tells us that we need to walk four units in the 𝑥-direction. And the second coordinate, the 𝑦-coordinate three, tells us that we need to move three units in the 𝑦-direction; that is three units parallel to the 𝑦-axis. And having done this, we’ve reached our point 𝑃. We could just have easily done it the other way around, first moving three units in the 𝑦-direction and then four units in the 𝑥-direction. It doesn’t matter which direction we go first.
Going back to our three-dimensional picture, let’s think about what the coordinates of 𝐴 tell us. Well, they tell us how to get to the point 𝐴 from the origin which I have marked. The first coordinate of the point 𝐴, the 𝑥-coordinate seven, tells us that we need to move seven units in the 𝑥-direction, in this case, along the 𝑥-axis. The second coordinate of the point 𝐴, the 𝑦-coordinate five, tells us that we need to move five units in the 𝑦-direction; that is five units parallel to the 𝑦-axis. The third coordinate is the new coordinate that we didn’t have in the two-dimensional case; it is the 𝑧-coordinate. And the fact that the 𝑧-coordinate is six tells us that we have to continue our journey by moving six units in the 𝑧-direction; that is six units parallel to the 𝑧-axis. So that is the interpretation of the coordinates of 𝐴.
I should just mention at this point that we’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, 𝐷𝐵 is parallel to the 𝑦-axis and 𝐵𝐴 is parallel to the 𝑧-axis. So now that we have clarified what the coordinates of a point in 3D space mean, we’re ready to determine the coordinates of 𝐵 and 𝐶.
First, let’s find the coordinates of 𝐵, which is the same as asking how to get to the point 𝐵 from the origin 𝑂. And we are going to need three coordinates: the first coordinate, the 𝑥-coordinate, tells us how far we have to go in the 𝑥-direction; the second, which is the 𝑦-coordinate, tells us how far we have to go in the 𝑦-direction; and the third, the 𝑧-coordinate, tells us how far we have to go in the 𝑧-direction. Well, if we look at our figure, we can see that we already have a path to 𝐵 from the origin; it’s through the point 𝐷. From the origin 𝑂, we first travel seven units in the 𝑥-direction to the point 𝐷. And once we are there, we just travel five units in the 𝑦-direction, and we get to the point 𝐵. So what are the coordinates of 𝐵? We travelled seven units in the 𝑥-direction, so the 𝑥-coordinate is seven and then five units in the 𝑦-direction, so the 𝑦-coordinate is five and then we were already at 𝐵, so we didn’t have to travel any distance in the 𝑧-direction, so our 𝑧-coordinate is zero.
And how about the point 𝐶? We haven’t already drawn the path to 𝐶 from the origin, but we notice again that we can go through 𝐷. And if you recall, we said that the length of the edge 𝐵𝐴 was six. And as we are in a cuboid or right-rectangular prism, the length of 𝐷𝐶 must be six also. So how do you get to 𝐶 from the origin? First, you travel seven units in the 𝑥-direction, so the 𝑥-coordinate is seven. We don’t have to move at all in the 𝑦-direction, and so the 𝑦-coordinate is zero. We just have to move six units in the 𝑧-direction, so the 𝑧-coordinate is six. So that’s our answer.
We’ve determined the coordinates of 𝐵 and 𝐶. You might like to check that going in a different route from the origin 𝑂 to the point 𝐵 or 𝐶 gives the same values of the coordinates as in the two-dimensional case.