In Euclidean space, the distance from a point to a plane is the distance between a given point and its orthogonal projection on the plane, the perpendicular distance to the nearest point on the plane.
It can be found starting with a change of variables that moves the origin to coincide with the given point then finding the point on the shifted plane that is closest to the origin. The resulting point has Cartesian coordinates :
- .
The distance between the origin and the point is .
Suppose we wish to find the nearest point on a plane to the point (), where the plane is given by . We define , , , and , to obtain as the plane expressed in terms of the transformed variables. Now the problem has become one of finding the nearest point on this plane to the origin, and its distance from the origin. The point on the plane in terms of the original coordinates can be found from this point using the above relationships between and , between and , and between and ; the distance in terms of the original coordinates is the same as the distance in terms of the revised coordinates.
The formula for the closest point to the origin may be expressed more succinctly using notation from linear algebra. The expression in the definition of a plane is a dot product , and the expression appearing in the solution is the squared norm . Thus, if is a given vector, the plane may be described as the set of vectors for which and the closest point on this plane to the origin is the vector
- .[1][2]
The Euclidean distance from the origin to the plane is the norm of this point,
- .
The vector equation for a hyperplane in -dimensional Euclidean space through a point with normal vector is or where .[3]
The corresponding Cartesian form is where .[3]
The closest point on this hyperplane to an arbitrary point is
and the distance from to the hyperplane is
- .[3]
Written in Cartesian form, the closest point is given by for where
- ,
and the distance from to the hyperplane is
- .
Thus in the point on a plane closest to an arbitrary point is given by
where
- ,
and the distance from the point to the plane is
- .
Cheney, Ward; Kincaid, David (2010). Linear Algebra: Theory and Applications. Jones & Bartlett Publishers. pp. 450, 451. ISBN 9781449613525.