# [Math]Finding the shortest distance between two lines in 3D

Keywords: Math, shortest distance between two lines

The distance between two lines in $$\mathbb R^3$$ is equal to the distance between parallel planes that contain these lines.

To find that distance first find the normal vector of those planes - it is the cross product of directional vectors of the given lines. For the normal vector of the form (A, B, C) equations representing the planes are:

$$ Ax + By + Cz + D_1 = 0 $$

$$ Ax + By + Cz + D_2 = 0 $$

Take coordinates of a point lying on the first line and solve for D1. Similarly for the second line and D2.

The distance we’re looking for is:

$$ d = \frac{\lvert D_1 - D_2 \rvert}{\sqrt{A^2 + B^2 + C^2}} $$

Origin:

Finding the shortest distance between two lines

https://math.stackexchange.com/a/429434/601445

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