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