Find the direction ratios of a vector perpendicular to the two li… - Vidyadip

Question

Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are $1, 3, 2$ and $–1, 1, 2$

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Answer

Let $L_1$ and $L_2$ be the two lines with direction ratios $1,3,2$ and $-1,1,2$ respectively.
Let the direction ratios of the vector perpendicular to $L_1$ and $L_2$ be $a, b, c.$
$\therefore a+3 b+2 c=0$
and $-a+b+2 c=0$
$\begin{array}{l}\therefore \frac{ a }{\left|\begin{array}{ll}3 & 2 \\1 & 2\end{array}\right|}=\frac{- b }{\left|\begin{array}{cc}1 & 2 \\-1 & 2\end{array}\right|}=\frac{ c }{\left|\begin{array}{cc}1 & 3 \\-1 & 1\end{array}\right|} \end{array}$
$\therefore \frac{ a }{6-2}=\frac{- b }{2+2}=\frac{ c }{1+3}$
$\therefore \frac{ a }{4}=\frac{- b }{4}=\frac{ c }{4}$
$\therefore$ The direction ratios of the vector are $4,-4,4$.

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