Question

A line passes through (3, -1, 2) and is perpendicular to
$\overline{r}=\overline{i}+\overline{j}-\overline{k}+\lambda (2\overline{i}-2\overline{j}+\overline{k})$ and
$\overline{r}=2\overline{i}+\overline{j}-3\overline{k}+\mu (\overline{i}-2\overline{j}+2\overline{k})$
find the equation.

Answer 1

$\because Thegivenlinesareparallelto(2\stackrel{\wedge }{i}-2\stackrel{\wedge }{j}+\stackrel{\wedge }{k})$

$and(\stackrel{\wedge }{i}-2\stackrel{\wedge }{j}+2\stackrel{\wedge }{k})respectively.$

$\therefore Equationofalineperpendiculartothegiven$

$twolinesmustalsobeperpendiculartothesetwo$

$linestowhichthegivenlinesareparallel.$

$Hence,$

$\overline{r^{{}^{\prime }}}=\left|\begin{array}{ccc}\stackrel{\wedge }{i}& \stackrel{\wedge }{j}& \stackrel{\wedge }{k}\\ 2& -2& 1\\ 1& -2& 2\end{array}\right|$

$⟹\stackrel{\wedge }{i}(-4+2)-\stackrel{\wedge }{j}(4-1)+\stackrel{\wedge }{k}(-4+2)$

$⟹-2\stackrel{\wedge }{i}-3\stackrel{\wedge }{j}-2\stackrel{\wedge }{k}$

$Wewantalinewhichpassesthrough(3,-1,2)$

$andparallelto-2\stackrel{\wedge }{i}-3\stackrel{\wedge }{j}-2\stackrel{\wedge }{k}$

$i.e,(3\stackrel{\wedge }{i}-\stackrel{\wedge }{j}+2\stackrel{\wedge }{k})+\lambda (-2\stackrel{\wedge }{i}-3\stackrel{\wedge }{j}-2\stackrel{\wedge }{k})$

$⟹(3\stackrel{\wedge }{i}-\stackrel{\wedge }{j}+2\stackrel{\wedge }{k})-\lambda (2\stackrel{\wedge }{i}+3\stackrel{\wedge }{j}+2\stackrel{\wedge }{k})$