The direction ratios of the line perpendicular to the lines having direction ratios $2,3,1$ and $1,2,1$ are
The direction ratios of the line perpendicular to the lines having direction ratios $2,3,1$
and $1,2,1$ are
$-2,1,1$
$1,1,1$
$1,-1,1$
$2,2,-2$
Solution
Let $\bar{a}$ and $\bar{b}$ be the vectors along the lines whose direction ratios are $2,3,1$ and $1,2,1$ respectively.
$\therefore \overline{\mathrm{a}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\hat{\mathrm{k}} \quad \text { and } \quad \overline{\mathrm{b}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$
A vector perpendicular to both $\bar{a}$ and $\bar{b}$ is given by,
$\bar{a} \times \bar{b}=\left|\begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
2 & 3 & 1 \\
1 & 2 & 1
\end{array}\right|=\hat{i}(3-2)-\hat{j}(2-1)+\hat{k}(4-3)=\hat{i}-\hat{j}+\hat{k}$
Hence d.r.s are $1,-1,1$