Another Frame

The coordinate frames used in math books are not usually placed in a real world scene, so the distances are not expressed in any particular unit. When you create your 3D world inside a computer you need to decide on the units you are using.

The picture shows our 3D world with a different coordinate frame placed in it. The points in the world are the same points as before. But now, with a different coordinate frame, they will have different representations.

In the previous frame, the point labeled P1 was given the representation:

Now, the distances along the axes in the new (green) frame are different. The column matrix that represents P1 in the new frame is roughly:

(I am not doing math, here. I'm looking at the picture and guessing distances. Try to do the same. The goal is not to calculate anything but to think about points in space.)

The same point has a different representation in each frame. When you represent a point with a column matrix you have to know what frame is being used.