Find the vector equation of the line passing through the point (2,3,-5) and making equal angles with the co-ordinate axes.

$\displaystyle \cos \alpha=\cos \beta=\cos \gamma=l \Rightarrow l^{2}+l^{2}+l^{2}=1 \Rightarrow 3 l^{2}=1, \therefore l=1 / \sqrt{3}$

Direction cosines of the line are $\displaystyle \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \Rightarrow$ the direction ratios are $\displaystyle 1,1,1$

∴ Vector equation of the line is: $\displaystyle \overrightarrow{\mathbf{r}}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}+\lambda(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})$