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Encyclopedia>Numbers and Quantities>Vectors>Three Dimensions>Distances>How to Find the Distance from a Point to a Line

Two orthogonal lines

In order to find the distance from a point to a line, you use the distance formula:

Formula

Distance Formula

The distance from a point to a line is

D = \frac{| \vec{Q P} \times \vec{v} |}{| \vec{v} |},

where $P$ is the point, $Q$ is a point on the line and $\vec{v}$ is a vector along the line.

Example 1

You have the line $x (t) = 1 + t$ , $y (t) = t$ , $z (t) = 2 - 2 t$ and the point $P = (1, 2, 3)$ . By inserting $t = 0$ into the parametric equation, you can see that the point $Q = (1, 0, 2)$ is on the line. That gives you

\begin{array}{l} \vec{Q P} \times \vec{v} & = \begin{matrix} (0, & 2, & 1) \\ \times \\ (1, & 1, & - 2) \end{matrix} \\ = (- 4 - 1, 1 - 0, 0 - 2) \\ = (- 5, 1, - 2), \\ D & = \frac{| (- 5, 1, - 2) |}{| (1, 1, - 2) |} \\ = \frac{\sqrt{25 + 1 + 4}}{\sqrt{1 + 1 + 4}} \\ = \frac{\sqrt{30}}{\sqrt{6}} \\ = \sqrt{5} . \end{array}

This means that the distance from $P$ to the line is $\sqrt{5}$ .

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