Three dimensional coordinate systems.

additional to x and y axes there is also a z axis in three dimensional coordinate systems.

Coordinate planes:

Coordinate planes are planes that pass through two axes.

These are the coordinate planes in three dimensions:

• xz-plane (y = 0)
• xy-plane (z = 0)
• yz-plane (x = 0)

Distance

Definition

The distance between the points $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$ is defined as $|PQ|=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}$

Spheres

Definition

A sphere with center P and radius $r\ge 0$ is defined as the set of all points $X\in {\mathbb{R}}^3$ with distance r to P.

• If P = (a,b,c) then (x,y,z) is on the sphere whenever $\sqrt{(x-a{)}^2+(y-b{)}^2+(z-c{)}^2}=r$ (the point (x,y,z) is at r distance from p so its in the sphere)

• This is equivalent to $(x-a{)}^2+(y-b{)}^2+(z-c{)}^2=r^2$