Shortest Distance from Origin - 3D

Let the shortest distance of the given plane from the origin be $d$ .

For a plane $a x + by + cz + d =0$ , the shortest distance from a point $(x_{0}, y_{0}, z_{0} )$ has the formula

$\text{Shortest Distance} = \frac{|ax_{0} + by_{0} + cz_{0} + d | }{\sqrt{ a^{2} + b^{2}+ c^{2} }}$

The coordinates of the origin are $(0,0,0)$ . Our plane is $4x - 3y + 12z - 78 = 0$ . We can plug these values in the formula to get the distance.

$\begin{aligned} d & = \frac{|4 \times 0 - 3 \times 0 + 12 \times 0 - 78 | }{\sqrt{ 4^{2} + (-3)^{2} + 12^{2}}} \\ & = \frac{|0 + 0 +0 - 78 | }{\sqrt{169}} \\ & = \frac{78}{13} = \boxed{6} ~~~~ _\square \end{aligned}$

The general form of this formula is proved in this note .