Question

Show that the vector $\overrightarrow{a}=\left(\hat{i}+\hat{j}+\hat{k}\right)$ is equally inclined to the coordinate axes.

Answer 1

Consider the problem

Let

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

and its 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 $(1,1,1)$. Therefore, direction cosines of $\overrightarrow{r}$

$\frac{1}{\sqrt{1^2+1^2+1^2}},\frac{1}{\sqrt{1^2+1^2+1^2}},\frac{1}{\sqrt{1^2+1^2+1^2}}$

$=\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$

then,

$\cos \alpha =\frac{1}{\sqrt{3}},\cos \beta =\frac{1}{\sqrt{3}}$ and $\cos \gamma =\frac{1}{\sqrt{3}}$

therefore,

$\Rightarrow \alpha =\beta =\gamma$

therefore, all are equally inclined with the coordinates axis.