The column matrix is a mathematical object that **represents** the point
using our chosen coordinate frame.
The column matrix is not the point.
The column matrix and the point are different things.
One is used to represent the other,
just as your name is used to represent you.

I hope this is obvious, but students who try to skip lightly through the mathematics of computer graphics sometimes are vague about this, to their eventual grief. Your computer graphics book will soon start representing the same point in several different ways. It is vital to keep separated the ideas of the geometrical point, and the several ways it might be represented.

**Note:** often people speak casually and say things like,
"the point (4, 12, -4)," as if the triple of numbers were
identical to the point.
This is just for convenience.
They really should say "the point that is represented in
our chosen coordinate system by (4, 12, -4)."

Are the distances measured along the coordinate axes in feet or in meters?