Distance from a point to a plane information

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 $ax+by+cz=d$ that is closest to the origin. The resulting point has Cartesian coordinates $(x,y,z)$ :

\displaystyle x={\frac {ad}{a^{2}+b^{2}+c^{2}}},\quad \quad \displaystyle y={\frac {bd}{a^{2}+b^{2}+c^{2}}},\quad \quad \displaystyle z={\frac {cd}{a^{2}+b^{2}+c^{2}}}

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The distance between the origin and the point $(x,y,z)$ is ${\sqrt {x^{2}+y^{2}+z^{2}}}$ .

Distance from a point to a plane information

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