Question

The direction cosines of the line which is perpendicular to the lines whose direction cosines are proportional to $(1,-1,2)$ and $(2,1,-1)$ are

• A. $\frac{-1}{\sqrt{35}},\frac{5}{\sqrt{35}},\frac{3}{\sqrt{35}}$
• B. $\frac{13}{\sqrt{35}},\frac{-1}{\sqrt{35}},\frac{1}{\sqrt{35}}$
• C. $\frac{2}{\sqrt{3}},\frac{5}{\sqrt{3}},\frac{7}{\sqrt{3}}$
• D. $\frac{3}{\sqrt{35}},\frac{5}{\sqrt{35}},\frac{7}{\sqrt{35}}$

Answer 1

The correct option is A

$\frac{-1}{\sqrt{35}},\frac{5}{\sqrt{35}},\frac{3}{\sqrt{35}}$

Explanation for the correct option:

We need to find the direction cosines of the line which is perpendicular to the lines whose direction cosines are proportional to $(1,-1,2)$ and $(2,1,-1)$

Let $l,m,n$ be the direction cosines of the required line - Since it is perpendicular to the lines whose direction cosines are proportional to $(1,-1,2)$and $(2,1,-1).$
Thus,
$l-m+2n=0$
$2l+m-n=0$
On solving above equations we get
$\frac{l}{-1}=\frac{m}{5}=\frac{n}{3}$
The direction ratios of the required lines are proportional to $(-1,5,3)$
Direction cosines are,
$\Rightarrow$$\frac{-1}{\sqrt{(-1{)}^2+5^2+3^2}},$$\frac{5}{\sqrt{(-1{)}^2+5^2+3^2}},$ $\frac{3}{\sqrt{(-1{)}^2+5^2+3^2}}$$=\frac{-1}{\sqrt{35}},$ $\frac{5}{\sqrt{35}}$$,\frac{3}{\sqrt{35}}$
Hence, the correct answer is option (A)