Question

If $(1,-2,-2)$ and $(0,2,1)$ are direction ratios of two lines, then the direction cosines of a perpendicular to both the lines are

• A. $(\frac{1}{3},-\frac{1}{3},\frac{2}{3})$
• B. $(\frac{2}{3},-\frac{1}{3},\frac{2}{3})$
• C. $(-\frac{2}{3},-\frac{1}{3},\frac{2}{3})$
• D. $(\frac{2}{\sqrt{14}},-\frac{1}{\sqrt{14}},\frac{3}{\sqrt{14}})$

Answer 1

Answer: (B)

$(\frac{2}{3},-\frac{1}{3},\frac{2}{3})$

Let a vector $\overrightarrow{A}=\hat{i}-2\hat{j}-2\hat{k}$
$\overrightarrow{B}=2\hat{j}+\hat{k}$
So, vector perpendicular to $\overrightarrow{A}\&\overrightarrow{B}$ will be
$=\overrightarrow{A}\times \overrightarrow{B}$$=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& -2& -2\\ 0& 2& 1\end{array}\right|$
$=2\hat{i}-1\hat{j}+2\hat{k}$

So, the direction ration of the perpendicular vector or line are (2,-1,2)
Direction cosines are $\frac{2}{\sqrt{2^2+{(-1)}^2+2^2}},\frac{-1}{\sqrt{2^2+{(-1)}^2+2^2}},\frac{2}{\sqrt{2^2+{(-1)}^2+2^2}}$
or $(\frac{2}{3},\frac{-1}{3},\frac{2}{3})$