If the direction ratios of the lines $L_1$ and $L_2$ are $2,-1,1$ and $3,-3,4$ respectively, then the…

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
  1. $\pm \frac{2}{\sqrt{6}}, \pm \frac{1}{\sqrt{6}}, \pm \frac{1}{\sqrt{6}}$
  2. $\pm \frac{1}{\sqrt{35}}, \pm \frac{5}{\sqrt{35}}, \pm \frac{3}{\sqrt{35}}$
  3. $\pm \frac{3}{\sqrt{34}}, \pm \frac{3}{\sqrt{34}}, \pm \frac{4}{\sqrt{34}}$
  4. $\pm \frac{1}{\sqrt{14}}, \pm \frac{2}{\sqrt{14}}, \pm \frac{3}{\sqrt{14}}$

Solution

Let direction cosines of line that is perpendicular to both $L_1$ and $L_2$ are $l, m, n$, then $ \begin{aligned} & 2 l-m+n=0 \text { and } 3 l-3 m+4 n=0 \\ & \Rightarrow \quad \frac{l}{-4+3}=\frac{m}{3-8}=\frac{n}{-6+3} \Rightarrow \frac{l}{-1}=\frac{m}{-5}=\frac{n}{-3} \\ & \Rightarrow \quad \frac{l}{l}=\frac{m}{5}=\frac{n}{3}= \pm \frac{\sqrt{l^2+m^2+n^2}}{\sqrt{1+25+9}} \\ & \Rightarrow \quad l, m, n= \pm \frac{1}{\sqrt{35}}, \pm \frac{5}{\sqrt{35}}, \pm \frac{3}{\sqrt{35}} . \end{aligned} $

Asked in: AP EAMCET 2018 (23 Apr Shift 2)

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