From Vector to Plane

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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 , with as the defining vector, , 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.


Contributed by: Ed Pegg Jr (March 2011)
Open content licensed under CC BY-NC-SA



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.

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