Distance from a Plane to a Point

Topics: Geometry - 3D Space - Plane - Point

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Let $\Pi$ be a plane and $q$ a point in ${\mathbb{R}}^3$:

$$\Pi =Ax+By+Cz+D=0$$

$$q=(q_1,q_2,q_3)$$

We can find the distance from $\Pi$ to $q$ by calculating:

$$d(q,\Pi )=\frac{|A{q}_1+B{q}_2+C{q}_3+D|}{\sqrt{A^2+B^2+C^2}}$$

Example

For example, let $\Pi =x-7y+5=0$ and $q=(1,-1,7)$.

We have that:

$$d(q,\Pi )=\frac{|1(1)-7(-1)+5|}{\sqrt{1^2+(-7{)}^2+0^2}}$$

$$=\frac{|13|}{\sqrt{50}}$$

$$=\frac{13}{5\sqrt{2}}$$