I'd like to know which all the different ways to mathematically represent a plane.
So far I only know one, if you have a vector normal to the plane, and a point on the plane, you can say ax + by + cz = d where a, b and c are the 3 numbers in the normal vector, then to get d, you simply substitute in the values of the point on the plane into x, y and z.
So that's how to get an equation for a plane using a normal and a point on the plane, are there any other ways? I'm also interested in what you can do with a plane, ie, distance to other points, whether it intersects other planes or something.
Let $a,b,c$ be any three non-colinear points in the plane. Let $\def\U{\mathbf{u}_b}\U = b-a$ and $\def\V{\mathbf{u}_c}\V = c-a$. Then the plane is the set of all the points of the form $$a + k_b\U +k_c\V$$ for real numbers $k_b, k_c$. The pairs $\langle k_b, k_c\rangle$ defined a coordinate system for the plane, in which $\langle 0,0\rangle$ represents point $a$, $\langle 1, 0\rangle$ represents point $b$, and $\langle 0, 1\rangle$ represents point $c$.