Question

If the direction ratios of the lines $L_{1}$ and $L_{2}$ are $2,-1,1$ and $3,-3,4$ respectively, then the direction cosines of a line that is perpendicular to both $L_{1}$ and $L_{2}$ are

• A. $\pm \frac{2}{\sqrt{6}}, \pm \frac{1}{\sqrt{6}}, \pm \frac{1}{\sqrt{6}}$
• B. $\pm \frac{1}{\sqrt{35}}, \pm \frac{5}{\sqrt{35}}, \pm \frac{3}{\sqrt{35}}$
• C. $\pm \frac{3}{\sqrt{34}}, \pm \frac{3}{\sqrt{34}}, \pm \frac{4}{\sqrt{34}}$
• D. $\pm \frac{1}{\sqrt{14}}, \pm \frac{2}{\sqrt{14}}, \pm \frac{3}{\sqrt{14}}$

Answer 1

Answer: (B)

Let direction cosines of line that is
perpendicular to both $L_{1}$ and $L_{2}$ are $l, m, n$, then
$2 l-m +n=0$ and $3 l-3 m+4 n=0$
$\Rightarrow \frac{l}{-4+3}=\frac{m}{3-8}=\frac{n}{-6+3}$
$\Rightarrow \frac{l}{-1}=\frac{m}{-5}=\frac{n}{-3}$
$\Rightarrow \frac{l}{1}=\frac{m}{5}=\frac{n}{3}=\pm \frac{\sqrt{l^{2}+m^{2}+n^{2}}}{\sqrt{1+25+9}}$
$\Rightarrow l, m, n=\pm \frac{1}{\sqrt{35}}, \pm \frac{5}{\sqrt{35}}, \pm \frac{3}{\sqrt{35}}$