Question

Find the magnitude of projection of vector $\left(2\hat{i}+3\hat{j}+\hat{k}\right)$ on a vector which is perpendicular to the plane containing vectors $\left(\hat{i}+\hat{j}+\hat{k}\right)$ and $\left(\hat{i}+2\hat{j}+\widehat{3k}\right)$.

• A. $\frac{\sqrt{3}}{\sqrt{2}}$
• B. $\frac{\sqrt{2}}{\sqrt{3}}$
• C. $\frac{4}{\sqrt{3}}$
• D. $\frac{2\sqrt{2}}{\sqrt{3}}$

Answer 1

The correct option is A

$\frac{\sqrt{3}}{\sqrt{2}}$

Explanation for the correct option:

Find out the normal vector perpendicular to plane containing given two vectors.

The vector perpendicular to both vectors containing $\left(\hat{i}+\hat{j}+\hat{k}\right)$ and $\left(\hat{i}+2\hat{j}+\widehat{3k}\right)$ is

$$\begin{gathered} =(\hat{i}+\hat{j}+\hat{k})\times (\hat{i}+2\hat{j}+3\hat{k}) \\ =(\hat{i}-2\hat{j}+\hat{k}) \end{gathered}$$

Therefore, the magnitude of the projection of vector $(2\hat{i}+\widehat{3j}+\hat{k})$ on $(\hat{i}-2\hat{j}+\hat{k})$ is

$$\begin{gathered} =\left|\frac{(2\hat{i}+3\hat{j}+\hat{k}).(\hat{i}-2\hat{j}+\hat{k})}{\sqrt{1^2+{(-2)}^2+1^2}}\right| \\ =\left|\frac{2-6+1}{\sqrt{6}}\right| \\ =\left|\frac{-3}{\sqrt{6}}\right| \\ =\frac{\sqrt{3}}{\sqrt{2}} \end{gathered}$$

Therefore, Option(A) is the correct answer.