Question

Find the minimum distance between the lines whose vector equations are:
$\overrightarrow{r}=\hat{i}+\hat{j}+l(2\hat{i}-\hat{j}+\hat{k})$ and $2\hat{i}+\hat{j}-\hat{k}+m(3\hat{i}-5\hat{j}+2\hat{k})$

Answer 1

$\overrightarrow{r}=\hat{i}+\hat{j}+l(2\hat{i}-\hat{j}+\hat{k})=\overrightarrow{a_1}+\lambda \overrightarrow{b_1}$ (Let)
$\overrightarrow{r}=2\hat{i}+\hat{j}-\hat{k}+m(3\hat{i}-5\hat{j}+2\hat{k})=\overrightarrow{a_2}+5\overrightarrow{b_2}$ (Let)
Now, $\overrightarrow{b_2}\times \overrightarrow{b_2}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& -1& 1\\ 3& -5& 2\end{array}\right|$
$=\hat{i}(-2+5)-\hat{j}(4-3)+\hat{k}(-10+3)=-3\hat{i}-\hat{j}-7\hat{k}$
$|\overrightarrow{b_2}\times \overrightarrow{b_2}|=\sqrt{{\left(3\right)}^2+{(-1)}^2+{(-)7}^2}=\sqrt{59}$
$\overrightarrow{a_2}-\overrightarrow{a_1}=2\hat{i}+\hat{j}-\hat{k}-\hat{i}-\hat{j}=\hat{i}-\hat{k}$
Shortest distance or minimum distance $=\left|\frac{(\overrightarrow{b_1}\times \overrightarrow{b_2})(\overrightarrow{a_2}-\overrightarrow{a_1})}{|\overrightarrow{b_1}\times \overrightarrow{b_2}|}\right|$
$=\left|\frac{3+7}{\sqrt{59}}\right|$
Minimum distance$=\frac{10}{\sqrt{59}}$