Search any question & find its solution
Question:
Answered & Verified by Expert
If $\overrightarrow{\mathrm{OA}}=2 \hat{i}-\hat{j}+\hat{k}, \overrightarrow{O B}=3 \hat{i}-\hat{k}$ and $\overrightarrow{O C}=2 \hat{j}+3 \hat{k}$ are the position vectors of the points $A, B$ and $C$, then a unit vector perpendicular to the plane containing $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ is
Options:
Solution:
1068 Upvotes
Verified Answer
The correct answer is:
$\frac{8 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}}{\sqrt{93}}$
$\begin{aligned}
& \text {} \overrightarrow{A B}=\overrightarrow{O B}-\overrightarrow{O A}=\hat{i}+\hat{j}-2 \hat{k} \\
& \overrightarrow{A C}=-2 \hat{i}+3 \hat{j}+2 \hat{k}
\end{aligned}$
$\because \overrightarrow{A B}$ and $\overrightarrow{A C}$ lies on the plane. Then $\overrightarrow{A B} \times \overrightarrow{A C}$ will be perpendicular to the plane.
Now, $\vec{n}=\overrightarrow{A B} \times \overrightarrow{A C}$
$=\left|\begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
1 & 1 & -2 \\
-2 & 3 & 2
\end{array}\right|=8 \hat{i}+2 \hat{j}+5 \hat{k}$
The unit vector perpendicular to the plane is
$\hat{n}=\frac{\vec{n}}{|\vec{n}|}=\frac{8 \hat{i}+2 \hat{j}+5 \hat{k}}{\sqrt{64+4+25}}=\frac{8 \hat{i}+2 \hat{j}+5 \hat{k}}{\sqrt{93}}$
& \text {} \overrightarrow{A B}=\overrightarrow{O B}-\overrightarrow{O A}=\hat{i}+\hat{j}-2 \hat{k} \\
& \overrightarrow{A C}=-2 \hat{i}+3 \hat{j}+2 \hat{k}
\end{aligned}$
$\because \overrightarrow{A B}$ and $\overrightarrow{A C}$ lies on the plane. Then $\overrightarrow{A B} \times \overrightarrow{A C}$ will be perpendicular to the plane.
Now, $\vec{n}=\overrightarrow{A B} \times \overrightarrow{A C}$
$=\left|\begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
1 & 1 & -2 \\
-2 & 3 & 2
\end{array}\right|=8 \hat{i}+2 \hat{j}+5 \hat{k}$
The unit vector perpendicular to the plane is
$\hat{n}=\frac{\vec{n}}{|\vec{n}|}=\frac{8 \hat{i}+2 \hat{j}+5 \hat{k}}{\sqrt{64+4+25}}=\frac{8 \hat{i}+2 \hat{j}+5 \hat{k}}{\sqrt{93}}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.