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.

Details

The vector is in standard position, starting at the origin. The plane passes through the tip of the vector.

Conversely, a plane determines the vector from the origin to the closest point to the plane from the origin.

External Links

Line (Wolfram MathWorld)

Normal Vector (Wolfram MathWorld)

Perpendicular (Wolfram MathWorld)

Vector (Wolfram MathWorld)

Vectors in 3D

Vector Norm (Wolfram MathWorld)

Permanent Citation

Ed Pegg Jr

"From Vector to Plane"
http://demonstrations.wolfram.com/FromVectorToPlane/
Wolfram Demonstrations Project
Published: March 7, 2011