Point Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard

In 3D space, describe the point

(2, -3, 5)

. How does adding a third coordinate change what a point represents compared to 2D?

Solution

1
Step 1: In 3D, a point is located by three coordinates $(x, y, z)$ .
2
Step 2: $(2, -3, 5)$ means: 2 units along x-axis, 3 units in the negative y-direction, 5 units up the z-axis.
3
Step 3: In 2D, a point needs 2 numbers (2 dimensions). In 3D, it needs 3 numbers (3 dimensions). Adding a coordinate extends our description into a new independent direction.

Answer

(2, -3, 5)

is a point in 3D space. A third coordinate locates the point along a new axis perpendicular to the 2D plane.

A point remains zero-dimensional (no size) regardless of how many coordinates describe it. The number of coordinates needed is the dimension of the space, not of the point itself. This generalises to