The direction cosines of a line which is perpendicular to lines whose direction ratios $3,-2,4$ and $1,3,-2$…
The direction cosines of a line which is perpendicular to lines whose direction ratios $3,-2,4$ and $1,3,-2$ are
- $\frac{-8}{\sqrt{285}}, \frac{-10}{\sqrt{285}}, \frac{11}{\sqrt{285}}$
- $\frac{-8}{\sqrt{285}}, \frac{10}{\sqrt{285}}, \frac{11}{\sqrt{285}}$
- $\frac{8}{\sqrt{285}}, \frac{10}{\sqrt{285}}, \frac{11}{\sqrt{285}}$
- $\frac{4}{\sqrt{297}}, \frac{5}{\sqrt{297}}, \frac{16}{\sqrt{297}}$
Solution
111.(B)
Let $\bar{a}=3 \hat{i}-2 \hat{j}+4 \hat{k}$ and $\bar{b}=1 \hat{i}+3 \hat{j}-2 \hat{k}$
$\begin{aligned}
\bar{a} \times \bar{b} &=\left|\begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
3 & -2 & 4 \\
1 & 3 & -2
\end{array}\right| \\
\bar{a} \times \bar{b} &=(4-12) \hat{i}-(-6-4) \hat{j}+(9+2) \hat{k} \\
&=-8 \hat{i}+10 \hat{j}+11 \hat{k} \\
|\bar{a} \times \bar{b}| &=\sqrt{(-8)^{2}+(10)^{2}+(11)^{2}}=\sqrt{285}
\end{aligned}$
Direction cosines are : $\frac{-8}{\sqrt{285}}, \frac{10}{\sqrt{285}}, \frac{11}{\sqrt{285}}$
Asked in: MHT CET 2020 (16 Oct Shift 1)
Practice more Three Dimensional Geometry questions on Aicharya