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 = |QP ×v| |v| ,

where P is the point, Q is a point on the line and v is a vector along the line.

Example 1

You have the line x (t) = 1 + t, y (t) = t, z (t) = 2 2t 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

QP ×v = (0, 2, 1 1) × ( 1, 1, 2) = (4 1, 1 0, 0 2) = (5, 1,2) , D = | (5, 1,2)| | (1, 1,2)| = 25 + 1 + 4 1 + 1 + 4 = 30 6 = 5.

This means that the distance from P to the line is 5.

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