Find the shortest distance between the lines whose vector equations are: vector (r) = (i + j) + λ(2i + j - k)

Question

Find the shortest distance between the lines whose vector equations are:

\(\vec{r}=(\hat{i}+\hat{j}) + \lambda (2\hat{i}-\hat{j}+\hat{k})\) and \(\vec{r}=(2\hat{i}+\hat{j}-\hat{k}​​) + \mu (3\hat{i}-5\hat{j}+2\hat{k}).\)

Answer 1

Comparing the given equations with equations

\(\vec{r}=\vec{a_1}+\lambda\, \vec{b_1}\) and \(\vec{r}=\vec{a_2}+\mu\, \vec{b_2}\)

we get \(\vec{a_1}=\hat{i} + \hat{j},\) \(\vec{b_1}=2\hat{i} - \hat{j} + \hat{k}\) and \(\vec{a_2}=2\hat{i} + \hat{j}-​​\hat{k},\) \(\vec{b_2}=3\hat{i} - 5\hat{j}+2​​\hat{k},\)

Hence, the shortest distance between the given lines is given by