The direction cosines of $\hat{i}+\hat{j}+\hat{k}$ are

Let $\overrightarrow{A}=\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k} \therefore A_x=1,A_y=1, A_z=1$and $A=\sqrt{A_x^2+A_y^2+A_z^2}=\sqrt{{\left(1\right)}^2+{\left(1\right)}^2+{\left(1\right)}^2}=\sqrt{3}$

let cos$\alpha$, cos$\beta$ and cos$\gamma$ be the direction cosines of $\overrightarrow{A}$. 

$\therefore \cos \alpha =\frac{A_x}{A}=\frac{1}{\sqrt{3}}, \cos \beta =\frac{A_y}{A}=\frac{1}{\sqrt{3}}, \cos \gamma =\frac{A_z}{A}=\frac{1}{\sqrt{3}}$