Question

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

Answer 1

Shortest distance $\to \left|\frac{\left[\right(b_1\times b_2).(a_2-a_1\left)\right]}{|b_1\times b_2|}\right|$

$b_1\times b_2=\left|\begin{array}{ccc}\hat{I}& \hat{J}& \hat{K}\\ 1& -1& 1\\ 2& 1& 2\end{array}\right|$

$=\hat{I}(-2-1)-\hat{J}(2-2)+\hat{K}(1+2)$

$=-3\hat{I}+3\hat{K}$

$|b_1\times b_2|=\sqrt{9+9}$

$=3\sqrt{2}$

$\left[\right(b_1\times b_2).(a_2-a_1\left)\right]=\left|\begin{array}{ccc}1& -1& 1\\ 2& 1& 2\\ 1& -1& -2\end{array}\right|$

$=1(-2+2)+1(-4-2)+1(-2-1)$

$=0+1(-6)+1(-3)$

$=-6-3=-9$

$SD=\left|\frac{-9}{3\sqrt{2}}\right|=\frac{3}{\sqrt{2}}$