# Finding the shortest distance between two lines

I know how to find the distance between a point and a line, not between two lines.

Find the shortest distance between the lines $(-1,1,4) + t(1,1,-1)$ and $(5,3,-3) + s(-2,0,1)$

Any help would be appreciated.

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Can you use differential calculus? –  DonAntonio Oct 11 '12 at 3:20
No, I cannot, I think I may have figured it out by doing the do product of the vectors t and s with the cross product of s and t, then subtracting them. Incidentally the shortest distance is 0, I believe. –  Unknown Oct 11 '12 at 3:25

## 2 Answers

Let $x_1$ and $y_1$ be 2 points on the line 1 and line 2 respectively. Form the difference vector $d=x_1-y_1$. Take another point $x_2$ on the line 1. Form the direction vector $x=x_1-x_2$. Project $d$ on to the direction vector $x$.

\begin{align} x_{parallel}= \frac{(d.x)}{||x||^2}x \end{align}

Now the norm of the following vector (the euclidean distance from the origin), will give you the required minimum distance.

\begin{align} x_{perp}= d-x_{parallel} \end{align}

(if they are not parallel, this will not work, instead it gives the shortest distance between the point $x_1$ and line 2.)

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The distance between two lines in $\Bbb R^3$ 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: $$d = \frac{|D_1 - D_2|}{\sqrt{A^2 + B^2 + C^2}}$$

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