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 $\hat{i}+\hat{j}+\hat{k}\&\hat{i}+2\hat{j}+3\hat{k}$ 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
$=\frac{\left|\right. \left(\right. 2 \hat{i} + 3 \hat{j} + \hat{k} \left.\right) \cdot \left(\right. \hat{i} - 2 \hat{j} + \hat{k} \left.\right) \left|\right.}{\left|\right. \hat{i} - 2 \hat{j} + \hat{k} \left|\right.}$
$=\frac{\left|\right. 2 - 6 + 1 \left|\right.}{\left|\right. \sqrt{6} \left|\right.}=\frac{3}{\sqrt{6}}=\sqrt{\frac{3}{2}}$