Question

Find the shortest distance between the lines
$\overrightarrow{r}=(4\hat{i}-\hat{j})+\lambda (\hat{i}+2\hat{j}-3\hat{k})$ and $\overrightarrow{r}=(\hat{i}-\hat{j}+2\hat{k})+\mu (\hat{i}+4\hat{j}-5\hat{k})$

Answer 1

Find the shortest distance between the lines

$\overline{r}=(4\hat{i}-\hat{j})+\lambda (\hat{i}+2\hat{j}-3\hat{k})$

$\hat{a}=(4,-1,0)$

$\hat{l}=(1,2,-3)$

$\hat{l}=(\hat{i}-\hat{j}+2\hat{k})+\mu (\hat{i}+4\hat{j}-5\hat{k})$

$\hat{b}=(1,-1,2)$

$\hat{m}=(1,4,--5)$

$\overline{l}\times \overline{m}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 2& -3\\ 1& 4& -5\end{array}\right|$

$=\hat{i}(-10+12)-\hat{j}(-5+3)+\hat{k}(4-2)$

$=(2,2,2)\ne 0$

$|\overline{l}\times \overline{m}|=\sqrt{4+4+4}$

$=\sqrt{12}>2\sqrt{3}$

$(\overline{b}-\overline{a}).(\overline{l}\times \overline{m})=(3,0,2)(2,2,2)$

$=6+4=10$

$distance=\frac{\left|\right(\overline{b}-\overline{a}).(\overline{l}\times \overline{m}\left)\right|}{|\overline{l}\times \overline{m}|}$

$=\frac{10}{2\sqrt{3}}=\frac{5}{\sqrt{3}}$