Video: Determining the Coordinates of a Point Drawn in the Cartesian Coordinate System

Determine the coordinates of point 𝐴.

03:14

Video Transcript

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 𝐡 lies in the π‘₯𝑦-plane. Let’s ignore the 𝑧-axis for a moment and forget that we’re in three-dimensional space and just focus on this π‘₯𝑦-plane. We can read off the π‘₯-coordinate three from the π‘₯-axis and the 𝑦-coordinate negative three from the 𝑦-axis.

So ignoring the third dimension, 𝐡 has coordinates three, negative three. You can think of these coordinates as instructions tell you how to get to 𝐡 from the origin. Starting at the origin, the π‘₯- coordinate tells us how far we have to move in the positive π‘₯-direction. So parallel to the π‘₯-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 𝐡.

This works fine in the π‘₯𝑦-plane where we just have two dimensions and two axes. We can get to 𝐡 just fine. But how do we get to 𝐴? We can’t do this by just moving parallel to the π‘₯- 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 π‘₯- and 𝑦-axes. We have to move three units in the 𝑧-direction. If we do this from 𝐡, we succeed in getting to 𝐴.

So putting this all together, to get to 𝐴, we have to move three units in the π‘₯-direction. That gives us our π‘₯-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 π‘₯-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 π‘₯𝑦-plane.

In our case, this was a point 𝐡. The π‘₯- and 𝑦-coordinates of 𝐴 in 3D space were just the π‘₯- and 𝑦-coordinates of 𝐡 in the 2D plane. The third 𝑧-coordinate told us how far 𝐴 was above 𝐡. Of course this would’ve been negative if 𝐴 were actually below 𝐡.

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