Question

Find a vector $\overrightarrow{r}$ of magnitude $3\sqrt{2}$ units which makes and angle $\frac{\pi }{4}$ and $\frac{\pi }{2}$ with $y$ and z-axis respectively

Answer 1

$\overrightarrow{r}=x\hat{i}+y\hat{j}+2\hat{k}\left|\overrightarrow{r}\right|=3\sqrt{2}\sqrt{x^2+y^2+z^2}=3\sqrt{2}{x}^2+y^2+z^2=18$

It makes $\frac{\pi }{4}$ with yours

So, $\overrightarrow{r}.\hat{j}=\left|\overrightarrow{r}\right|\cos \frac{\pi }{4}y=\frac{3\sqrt{2}}{\sqrt{2}}=3$

It makes $\frac{\pi }{2}$ with $z-axis$

So,$\overrightarrow{r}.\hat{k}=\left|\overrightarrow{r}\right|\cos \frac{\pi }{2}z=0$

So, $x^2+3^2+0^2=18x^2+9=18x^2=9x=\pm 3$

So, $\overrightarrow{r}=3\hat{i}+3\hat{j}/-3\hat{i}+3\hat{j}$