Question

Show that the vector $\hat{i}+\hat{j}+\hat{k}$ is equally inclined with the axes OX, OY and OZ.

Answer 1

Let $\overrightarrow{r}=\hat{i}+\hat{j}+\hat{k}$ and it make an angle $\alpha ,\beta ,\gamma$ with $OX,OY,OZ$ respectively.Then its direction cosines are $\cos \alpha ,\cos \beta$ and $\cos \gamma$. So, Direction ratios of $\overrightarrow{r}=\hat{i}+\hat{j}+\hat{k}$ are proportional $\left(1,1,1\right)$. Therefore, direction cosines of $\overrightarrow{r}$ are
$\frac{1}{\sqrt{1^2+1^2+1^2}},\frac{1}{\sqrt{1^2+1^2+1^2}},\frac{1}{\sqrt{1^2+1^2+1^2}}$ or, $\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$.
Thus,
$\cos \alpha =\frac{1}{\sqrt{3}},\cos \beta =\frac{1}{\sqrt{3}}$ and $\cos \gamma =\frac{1}{\sqrt{3}}$.
$\Rightarrow \alpha =\beta =\gamma$.
Hence, all are equally inclined with the coordinate axis.