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 2 x-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}} $
$\therefore \vec{r}=\frac{1}{\sqrt{2}} \hat{i}-\frac{1}{\sqrt{2}} \hat{k}$