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

Answer 1

Answer: (B)

Vector perpendicular to plane containing the vectors $\hat{i}+\hat{j}+\hat{k}$ and $\hat{i}+2\hat{j}+3\hat{k}$ is parallel to vector
$=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 1& 1\\ 1& 2& 3\end{array}\right|=\hat{i}-2\hat{j}+\hat{k}$
$\therefore$ Required magnitude of projection
$=\frac{|\left(2\hat{i}+3\hat{j}+\hat{k}\right).\left(\hat{i}-2\hat{j}+\hat{k}\right)|}{|\hat{i}-2\hat{j}+\hat{k}|}$
$=\frac{|2-6+1|}{|\sqrt{6}|}=\frac{3}{\sqrt{6}}=\sqrt{\frac{3}{2}}$