Q78 / 120 · VITEEE 2018 · Mathematics
A unit vector perpendicular to the plane formed by the points $(1,0,1),(0,2,2)$ and $(3,3,0)$ is
- A$\frac{1}{5 \sqrt{3}}(5 \hat{\mathrm{i}}-\hat{\mathrm{j}}-7 \hat{\mathrm{k}})$
- Correct answerB$\frac{1}{5 \sqrt{3}}(5 \hat{\mathrm{i}}-\hat{\mathrm{j}}+7 \hat{\mathrm{k}})$
- C$\frac{1}{5 \sqrt{3}}(5 \hat{\mathrm{i}}+\hat{\mathrm{j}}+7 \hat{\mathrm{k}})$
- DNone of these
Solution
Let $\mathrm{A}(1,0,1), \mathrm{B}(0,2,2)$ and $\mathrm{C}(3,3,0)$ be the given points, then
$$
\begin{array}{l}
\overrightarrow{\mathrm{AB}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{BC}}=3 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}} \\
\overrightarrow{\mathrm{AB}} \times \overrightarrow{\mathrm{BC}}=\left|\begin{array}{ccc}
\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\
-1 & 2 & 1 \\
3 & 1 & -2
\end{array}\right|=-5 \hat{\mathrm{i}}+\hat{\mathrm{j}}-7 \hat{\mathrm{k}} \\
\therefore \text { unit vector } \perp \text { to the plane: } \mathrm{ABC} \\
=\pm \frac{1}{5 \sqrt{3}}(-5 \hat{\mathrm{i}}+\hat{\mathrm{j}}-7 \hat{\mathrm{k}})
\end{array}
$$
Hence, the correct answer is option: 'B'.