Question

Direction cosines of the normal to the plane containing the lines $\frac{x}{2}=\frac{y}{3}=\frac{z}{5}$ and $\frac{x-1}{2}=\frac{y-1}{3}=\frac{z-1}{5}$ are

• A. $\frac{2}{\sqrt{14}},\frac{-3}{\sqrt{14}},\frac{5}{\sqrt{14}}$
• B. $\frac{2}{\sqrt{14}},\frac{-3}{\sqrt{14}},\frac{1}{\sqrt{14}}$
• C. $\frac{2}{\sqrt{14}},\frac{-1}{\sqrt{14}},\frac{1}{\sqrt{14}}$
• D. $\frac{3}{\sqrt{13}},\frac{-2}{\sqrt{13}},1$

Answer 1

Answer: (B)

The normal vector to the plane $=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 1& 1\\ 2& 3& 5\end{array}\right|$
$=2\hat{i}-3\hat{j}+\hat{k}$
Hence, direction cosines are $\left(\frac{2}{\sqrt{14}},\frac{-3}{\sqrt{14}},\frac{1}{\sqrt{14}}\right)$