Distance between two straight lines in space

The distance between two straight lines $r$ and $r^{'}$ , $d (r, r^{'})$ , is the minimal distance between any point of $r$ and any other point of $r^{'}$ .

If the straight lines coincide or are secant, the distance between them is zero, $d (r, r^{'}) = 0$ .
If the straight lines are parallel, the distance between them can be calculated at any point of one of the lines, $P \in r$ or $P^{'} \in r^{'}$ , and finding the distance to the other straight line: $d (r, r^{'}) = d (P, r^{'}) = d (r, P^{'})$
If the straight lines cross, the following general formula is deduced to calculate the distance between them:

We take a point $A$ belonging to $r$ and another point $A^{'}$ belonging to $r^{'}$ . Let $\vec{v}$ and ${\vec{v}}^{'}$ be the governing vectors of $r$ and $r^{'}$ . We join the points $A$ and $A^{'}$ . The volume of the parallelepiped determined by $\vec{A A^{'}}$ , $\vec{v}$ and ${\vec{v}}^{'}$ , is the absolute value of the mixed product of these vectors: $v_{p} = | [\vec{A A^{'}}, \vec{v}, {\vec{v}}^{'}] |$

On the other hand we can also calculate this volume by multiplying the area of the base and the height: $v_{p} = | \vec{v} \times {\vec{v}}^{'} | d (r, r^{'}) |$

Therefore: $d (r, r^{'}) = \frac{| [\vec{A A^{'}}, \vec{v}, {\vec{v}}^{'}] |}{| \vec{v} \times {\vec{v}}^{'} |}$

Example

We are going to calculate the distance between the straight lines: $r : x - 2 = \frac{y + 3}{2} = z r^{'} : x = y = z$

First we determine its relative position. To do it we must write the implicit equations of the straight line: $r : {\begin{cases} 2 x - y - 7 = 0 \\ x - z - 2 = 0 \end{cases} r^{'} : {\begin{cases} x - y = 0 \\ x - z = 0 \end{cases}$

And we calculate the rank of the matrix of the resulting systems of equations: $| M^{'} | = | \begin{matrix} 2 & - 1 & 0 & 7 \\ 1 & 0 & - 1 & 2 \\ 1 & - 1 & 0 & 0 \\ 1 & 0 & - 1 & 0 \end{matrix} | = 2 \neq 0$

Therefore $rank (M^{'}) = 4$ and the two straight lines intersect. Now we must find a point and the governing vector of each line.

For the straight line $r$ : $A = (2, - 3, 0)$ and $\vec{v} = (1, 2, 1)$ .

For the straight line $r^{'}$ : $A^{'} = (0, 0, 0)$ and $\vec{v} = (1, 1, 1)$ .

So we have: $\vec{A A^{'}} = (- 2, 3, 0)$

$\begin{aligned} | \vec{v} \times {\vec{v}}^{'} | = & | | \begin{array}{c} \vec{i} & \vec{j} & \vec{k} \\ 1 & 2 & 1 \\ 1 & 1 & 1 \end{array} | | = | 2 \vec{i} + \vec{j} + \vec{k} - 2 \vec{k} - \vec{j} - \vec{i} | = | \vec{i} - \vec{k} | \\ = & | (1, 0, - 1) | = \sqrt{1^{2} + 0^{2} + (- 1)^{2}} = \sqrt{2} \end{aligned}$

$[\vec{A A^{'}}, \vec{v}, {\vec{v}}^{'}] = | \begin{matrix} - 2 & 3 & 0 \\ 1 & 2 & 1 \\ 1 & 1 & 1 \end{matrix} | = - 4 + 3 + 2 - 3 = - 2$

Finally: $d (r, r^{'}) = \frac{| [\vec{A A^{'}}, \vec{v}, {\vec{v}}^{'}] |}{| \vec{v} \times {\vec{v}}^{'} |} = \frac{| - 2 |}{\sqrt{2}} = \sqrt{2}$