Question

The unit vector in ZOX plane, making angles $45^{\circ }$ and $60^{\circ }$ respectively with $\vec{\alpha }=2\hat{i}+2\hat{j}-\hat{k}$ and $\vec{\beta }=\hat{j}-\hat{k}$ is

• A. $\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{j}$
• B. $\frac{1}{\sqrt{2}}\hat{i}-\frac{1}{\sqrt{2}}\hat{k}$
• C. $\frac{1}{\sqrt{2}}\hat{i}-\frac{1}{\sqrt{2}}\hat{j}$
• D. $\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{k}$

Answer 1

Answer: (B)

Let the vector be $\vec{r}=x\hat{i}+z\hat{k}$
$\Rightarrow |\vec{r}|=1$
$\vec{r}\cdot \vec{\alpha }=|\vec{r}||\vec{\alpha }|\cos 45^{\circ }$
$\therefore 2x-z=\frac{3}{\sqrt{2}}$
$\vec{r}\cdot \vec{\beta }=|\vec{r}||\vec{\beta }|\cos 60^{\circ }$
$z=-\frac{1}{\sqrt{2}}$
$\therefore x=\frac{1}{\sqrt{2}}$
$\therefore \vec{r}=\frac{1}{\sqrt{2}}\hat{i}-\frac{1}{\sqrt{2}}\hat{k}$