Question

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

Answer 1

Let \[\vec{r} = \hat{i} + \hat{j} + \hat{k}\] and it make an angle \[\vec{r} = \hat{i} + \hat{j} + \hat{k}\] with OX, OY, OZ respectively.
Then its direction cosines are \[\cos \alpha, \cos \beta\] and \[\cos \gamma\].  
So, Direction ratios of \[\vec{r} = \hat{i} + \hat{j} + \hat{k}\] are proportional 1, 1, 1.
Therefore, direction cosines of \[\vec{r}\] 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.