3-D coordinate system

3-D Coordinate System
In the 3-D Coordinate System, also denoted as $\mathbb{R}^3$, we have 3 axis $(x,y,z)$. We draw the axis' in the graph like this:

Points are written in the form $P=(x,y,z)$

Planes
The $xy$-plane corresponds to all the points at which $z=0$.
The $xz$-plane corresponds to all the points at which $y=0$.
The $yz$-plane corresponds to all the points at which $x=0$.

These planes are sometimes called "coordinate planes".

Projection of a Point
When a point $P=(x_1,y_1,z_1)$ gets projected onto a plane, then the point will be on the plane.
For example, the projection of point $P=(x_1,y_1,z_1)$ onto the $xy$-plane will become $(x_1,y_1,0)$.

Distance of Two Points To find the distance between two points in $\mathbb{R}^3$, we use the formula:

$D = \sqrt{(x_2-x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

Other General Equations in 3D
Here are the general types of equations you might see in the 3-D Coordinate system:

1. Cylinder: $(x-a)^2 + (y-b)^2 = r^2$
   
   Points are written in the form $P=(x,y,z)$
2. Sphere: $(x-a)^2+(y-b)^2+(z-c)^2=r^2$
   
   Points are written in the form $P=(x,y,z)$
3. Plane: $ax+by+cz=d$