Question

A vector $\overrightarrow{r}$ makes angles of ${45}^o,{60}^o,{120}^o$ with $X,Y$ and $Z-axis$ respectively. If magnitude of $\overrightarrow{r}$ is $2$ units, then find $\overrightarrow{r}$.

Answer 1

$\because \frac{\overrightarrow{r}}{|\overrightarrow{r}|}=l\hat{i}+m\hat{j}+n\hat{k}$
Given: $|\overrightarrow{r}|=2$
and $\alpha ={45}^o,\beta ={60}^o,\gamma ={120}^o$
$\overrightarrow{r}=2[\cos {45}^o\hat{i}+\cos {60}^o\hat{j}+\cos {120}^o\hat{k}]$
$\Rightarrow \overrightarrow{r}=2[\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{2}\hat{j}-\frac{1}{2}\hat{k}]$
$\Rightarrow \overrightarrow{r}=\frac{2}{2}[\sqrt{2}\hat{i}+\hat{j}-\hat{k}]$
$\sqrt{2}\hat{i}+\hat{j}-\hat{k}$