Question

The magnitude of the projection of the vector $2\hat{i}+3\hat{j}+\hat{k}$ on the vector perpendicular to the plane containing the vectors $\hat{i}+\hat{j}+\hat{k}$ and $\hat{i}+2\hat{j}+3\hat{k},$ is:

• A. $3\sqrt{6}$
• B. $\frac{\sqrt{3}}{2}$
• C. $\sqrt{6}$
• D. $\sqrt{\frac{3}{2}}$

Answer 1

The correct option is D $\sqrt{\frac{3}{2}}$

The vector perpendicular to the given vector is
$\overrightarrow{a}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 1& 1\\ 1& 2& 3\end{array}\right|$

$\Rightarrow \overrightarrow{a}=\hat{i}-2\hat{j}+\hat{k}$
Projection of $\overrightarrow{b}$ on $\overrightarrow{a}=\frac{|\overrightarrow{a}\cdot \overrightarrow{b}|}{|\overrightarrow{a}|}$
$=\frac{|(\hat{i}-2\hat{j}+\hat{k})\cdot (2\hat{i}+3\hat{j}+\hat{k})|}{\sqrt{6}}$
$=\frac{|2-6+1|}{\sqrt{6}}$
$=\sqrt{\frac{3}{2}}$