Question

A vector $\overrightarrow{r}$ is inclined at equal angles to the three axes. If the magnitude of $\overrightarrow{r}$ is $2\sqrt{3}$ units, find $\overrightarrow{r}$.

Answer 1

Since $\overrightarrow{r}$ is inclined equal angles to axes

$\therefore$ $\alpha =\beta =y$

$3{\cos }^2\alpha =1$

$\therefore$ $3{\cos }^{}\alpha ==\pm \frac{1}{\sqrt{3}}$

Now componenets in each direction are equal

Let $\overrightarrow{r}=a\hat{i}+a\hat{j}+a\hat{k}$

$\therefore \left|\overrightarrow{r}\right|=\sqrt{a^2+a^2+a^2}=\sqrt{3}a$

$\sqrt{3}a=2\sqrt{3}\Rightarrow a=2$

$\overrightarrow{r}=2\hat{i}+2\hat{j}+2\hat{k}$