Shortest Distance Between Two Lines

The formula

For two lines $\vec{r}=\vec{a_1}+\lambda\vec{b_1}$ and $\vec{r}=\vec{a_2}+\mu\vec{b_2}$that don't intersect, the shortest distance between them is:

Worked example

Find the shortest distance between $\vec{r}=\hat{i}+\hat{j}+\lambda(2\hat{i}-\hat{j}+\hat{k})$ and $\vec{r}=2\hat{i}+\hat{j}-\hat{k}+\mu(3\hat{i}-5\hat{j}+2\hat{k})$.

Solution: Here $\vec{a_2}-\vec{a_1} = \hat{i}-\hat{k}$, $\vec{b_1}=(2,-1,1)$, $\vec{b_2}=(3,-5,2)$.

A second worked example

Find the shortest distance between $\vec{r}=\hat{i}+2\hat{j}+\hat{k}+\lambda(\hat{i}-\hat{j}+\hat{k})$ and $\vec{r}=2\hat{i}-\hat{j}-\hat{k}+\mu(2\hat{i}+\hat{j}+2\hat{k})$.

Solution: $\vec{a_2}-\vec{a_1} = \hat{i}-3\hat{j}-2\hat{k}$, $\vec{b_1}=(1,-1,1)$, $\vec{b_2}=(2,1,2)$.