Question

The direction cosines of the normal to the plane containing the lines having direction ratios $1,2,1$ and $4,5,-3$ are

• A. $\frac{-11}{\sqrt{179}}, \frac{7}{\sqrt{179}}, \frac{-3}{\sqrt{179}}$
• B. $\frac{1}{\sqrt{2}}, 0, \frac{-1}{\sqrt{2}}$
• C. $\frac{5}{\sqrt{41}}, \frac{-4}{\sqrt{41}}, 0$
• D. $\frac{2}{\sqrt{5}}, \frac{-1}{\sqrt{5}}, 0$

Answer 1

Answer: (A)

Let $b_{1}=\hat{i}+2 \hat{j}+\hat{k}$ and $b_{2}=4 \hat{i}+5 \hat{j}-3 \hat{k}$
Normal vector to plane
$b_{1} \times b_{2}=\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 1 \\ 4 & 5 & -3\end{vmatrix}$
$=\hat{i}(-6-5)-\hat{j}(-3-4)+\hat{k}(5-8)$
$=-11 \hat{i}+7 \hat{j}-3 \hat{k}$
$\therefore $ Direction cosines of normal to the plane
$=\frac{-11}{\sqrt{(-11)^{2}+7^{2}+(-3)^{2}}}, \frac{7}{\sqrt{(-11)^{2}+7^{2}+(-3)^{2}}}$
$\frac{-3}{\sqrt{(-11)^{2}+7^{2}+(-3)^{2}}}=\frac{-11}{\sqrt{179}}, \frac{7}{\sqrt{179}}, \frac{-3}{\sqrt{179}}$