Determine the coordinates of point 𝐴.
Hopefully, we know how to find the coordinates of a point in two dimensions, so on the plane. Our point 𝐴 however, like us, lives inside three-dimensional space. How do you find its coordinates?
We can use what we know about coordinates in the plane to help us. The point 𝐵 lies in the 𝑥𝑦-plane. Let’s ignore the 𝑧-axis for a moment and forget that we’re in three-dimensional space and just focus on this 𝑥𝑦-plane. We can read off the 𝑥-coordinate three from the 𝑥-axis and the 𝑦-coordinate negative three from the 𝑦-axis.
So ignoring the third dimension, 𝐵 has coordinates three, negative three. You can think of these coordinates as instructions tell you how to get to 𝐵 from the origin. Starting at the origin, the 𝑥- coordinate tells us how far we have to move in the positive 𝑥-direction. So parallel to the 𝑥-axis, we have to move three units.
And the 𝑦-coordinate tells us how far we have to move in the positive 𝑦-direction. So parallel to the 𝑦-axis, we have to move in negative three units in the positive 𝑦-direction. So that means moving three units in the other direction. And we see that if we do this we do indeed get to 𝐵.
This works fine in the 𝑥𝑦-plane where we just have two dimensions and two axes. We can get to 𝐵 just fine. But how do we get to 𝐴? We can’t do this by just moving parallel to the 𝑥- and 𝑦-axes. We have to move in the 𝑧-direction as well. How many units do we have to move in the 𝑧-direction?
We can read off the value from the 𝑧-axis just as we did from the 𝑥- and 𝑦-axes. We have to move three units in the 𝑧-direction. If we do this from 𝐵, we succeed in getting to 𝐴.
So putting this all together, to get to 𝐴, we have to move three units in the 𝑥-direction. That gives us our 𝑥-coordinate, three. Then we have to move negative three units in the 𝑦- direction. That gives us our 𝑦-coordinate, negative three. And finally we have to move three units in this new 𝑧-direction, giving us a 𝑧-coordinate of three.
We can write our answer like this: 𝐴 has coordinates three, negative three, three. As we’re working in three dimensions, there are three coordinates: the 𝑥-coordinate, 𝑦-coordinate, and the new 𝑧-coordinate. A good way to find the coordinates of a point in 3D space is to look for the point directly below it in the 𝑥𝑦-plane.
In our case, this was a point 𝐵. The 𝑥- and 𝑦-coordinates of 𝐴 in 3D space were just the 𝑥- and 𝑦-coordinates of 𝐵 in the 2D plane. The third 𝑧-coordinate told us how far 𝐴 was above 𝐵. Of course this would’ve been negative if 𝐴 were actually below 𝐵.