Question

A non-zero vector $\vec{a}$ is such that its projections along vectors $\frac{\hat{i}+\hat{j}}{\sqrt{2}},\frac{-\hat{i}+\hat{j}}{\sqrt{2}}$ and $\hat{k}$ are equal, then unit vector along $\vec{a}$ is

• A. $\frac{\sqrt{2}\hat{j}-\hat{k}}{\sqrt{3}}$
• B. $\frac{\hat{j}-\sqrt{2}\hat{k}}{\sqrt{3}}$
• C. $\frac{\sqrt{2}}{\sqrt{3}}\hat{j}+\frac{\hat{k}}{\sqrt{3}}$
• D. $\frac{\hat{j}-\hat{k}}{\sqrt{2}}$

Answer 1

Answer: (C)

Let the projection be $x$, then
$\vec{a}=\frac{x(\hat{i}+\hat{j})}{\sqrt{2}}+\frac{x(-\hat{i}+\hat{j})}{\sqrt{2}}+x\hat{k}$
$\therefore \vec{a}=\frac{2x\hat{j}}{\sqrt{2}}+x\hat{k}$ or $\hat{a}=\frac{\sqrt{2}}{\sqrt{3}}\hat{j}+\frac{\hat{k}}{\sqrt{3}}$