Question

The direction cosine of a line which is perpendicular to both the lines whose direction ratios are $1,2,2$ and $0,2,1$ are

• A. $\frac{-2}{3},\frac{1}{3},\frac{2}{3}$
• B. $\frac{2}{3},\frac{-1}{3},\frac{2}{3}$
• C. $\frac{2}{3},\frac{1}{3},\frac{-2}{3}$
• D. $\frac{2}{3},\frac{-1}{3},\frac{-2}{3}$

Answer 1

The correct option is B $\frac{2}{3},\frac{-1}{3},\frac{2}{3}$

Since we need the direction cosines of a line perpendicular to both the lines, having direction ratios $1,2,2$ and $0,2,1$; we first find the direction ratios of the required line by finding the cross product of the given ratios.

It would result in $(\hat{i}+2\hat{j}+2\hat{k})\times (2\hat{j}+\hat{k})$, i.e. $2\hat{k}-\hat{j}+0-2\hat{i}+4\hat{i}+0$

$=2\hat{i}-\hat{j}+2\hat{k}$

Thus, the direction ratios become $2,-1,2$

And the corresponding direction cosines become $\frac{2}{3},\frac{-1}{3},\frac{2}{3}$