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
$ = \begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\ 1&1&1\\ 1&2&3\end{vmatrix} = \hat{i} - 2\hat{j} + \hat{k} $
$ \therefore $ Required magnitude of projection
$= \frac{\left|\left(2\hat{i} + 3\hat{j} +\hat{k}\right).\left(\hat{i} -2 \hat{j} + \hat{k}\right)\right|}{\left|\hat{i} - 2\hat{j} + \hat{k}\right|} $
$= \frac{\left|2-6+1\right|}{\left|\sqrt{6}\right|} = \frac{3}{\sqrt{6}}= \sqrt{\frac{3}{2}} $