Question

A vector $\overrightarrow{r}$ is inclined at equal acute angles to $x$-axis, $y$-axis and $z$-aixs. If $|\overrightarrow{r}|=6$ units find $\overrightarrow{r}$ .

Answer 1

$\left|\overrightarrow{r}\right|=6unit$

Let the vector be $\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}$

Subtends equal angles with $x,y$ and $z$ axes.

$\Rightarrow \frac{x}{\sqrt{x^2+y^2+z^2}}=\frac{y}{\sqrt{y^2+x^2+z^2}}=\frac{y}{\sqrt{x^2+y^2+z^2}}$

$\Rightarrow x=y=z$

But we know $\left|\overrightarrow{r}\right|=6$

$\Rightarrow \sqrt{x^2+y^2+z^2}=6$

$\Rightarrow x^2+x^2+x^2=36$

$\Rightarrow 3x^2=36$

$\Rightarrow x^2=12$

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

$\therefore \overrightarrow{r}=2\sqrt{3}(\hat{i}+\hat{j}+\hat{k})$