### Video Transcript

Determine the coordinates of point π΄.

Hopefully, we know how to find the coordinates of a point in two dimensions, so on the plane. Our point π΄ 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 π΅ lies in the π₯π¦-plane. Letβs ignore the π§-axis for a moment and forget that weβre in three-dimensional space and just focus on this π₯π¦-plane. We can read off the π₯-coordinate three from the π₯-axis and the π¦-coordinate negative three from the π¦-axis.

So ignoring the third dimension, π΅ has coordinates three, negative three. You can think of these coordinates as instructions tell you how to get to π΅ 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 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 in negative three units in the positive π¦-direction. So that means moving three units in the other direction. And we see that if we do this we do indeed get to π΅.

This works fine in the π₯π¦-plane where we just have two dimensions and two axes. We can get to π΅ just fine. But how do we get to π΄? We canβt do this by just moving parallel to the π₯- and π¦-axes. We have to move in the π§-direction as well. How many units do we have to move in the π§-direction?

We can read off the value from the π§-axis just as we did from the π₯- and π¦-axes. We have to move three units in the π§-direction. If we do this from π΅, we succeed in getting to π΄.

So putting this all together, to get to π΄, 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 π¦- direction. That gives us our π¦-coordinate, negative three. And finally we have to move three units in this new π§-direction, giving us a π§-coordinate of three.

We can write our answer like this: π΄ has coordinates three, negative three, three. As weβre working in three dimensions, there are three coordinates: the π₯-coordinate, π¦-coordinate, and the new π§-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₯π¦-plane.

In our case, this was a point π΅. The π₯- and π¦-coordinates of π΄ in 3D space were just the π₯- and π¦-coordinates of π΅ in the 2D plane. The third π§-coordinate told us how far π΄ was above π΅. Of course this wouldβve been negative if π΄ were actually below π΅.