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
Suppose we wish to find the nearest point on a plane to the point (
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
; 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.
in the definition of a plane is a dot product
appearing in the solution is the squared norm
and the closest point on this plane to the origin is the vector The Euclidean distance from the origin to the plane is the norm of this point, In either the coordinate or vector formulations, one may verify that the given point lies on the given plane by plugging the point into the equation of the plane.
To see that it is the closest point to the origin on the plane, observe that
is a scalar multiple of the vector
itself, then the line segments from the origin to
form a right triangle, and by the Pythagorean theorem the distance from the origin to
must be a positive number, this distance is greater than
[2] Alternatively, it is possible to rewrite the equation of the plane using dot products with
in place of the original dot product with
(because these two vectors are scalar multiples of each other) after which the fact that
is the closest point becomes an immediate consequence of the Cauchy–Schwarz inequality.
[1] The vector equation for a hyperplane in
-dimensional Euclidean space
[3] The corresponding Cartesian form is
[3] The closest point on this hyperplane to an arbitrary point
to the hyperplane is Written in Cartesian form, the closest point is given by
closest to an arbitrary point
given by where and the distance from the point to the plane is