From Vector to Plane

A

1

B

2

C

3

The vector (1, 2, 3) corresponds to the plane

x+2y+3z14.

Any nonzero vector defines a unique plane in 3D. Except for planes through the origin, every plane is defined by a unique vector. This vector is normal (perpendicular) to the plane. In the equation of the plane

Ax+By+Cz=D

, with

(A,B,C)

as the defining vector,

D=

2

A

+

2

B

+

2

C

, which is the square of the norm (length) of the vector.

A vector norm is a length. A normal vector is perpendicular to a plane or line.