The direction cosines of a vector $\overline{A}$ are $\cos \alpha =\frac{4}{5\sqrt{2}},\cos \beta =\frac{1}{\sqrt{2}},\cos \gamma =\frac{3}{5\sqrt{2}}$ then the vector $\overline{A}$ is 

If $\overline{A}$ is a vector then $\left|\overline{A}\right|$ is its magnitude. 

 $\cos \alpha =\frac{A_x}{\left|\overline{A}\right|},\text{\hspace{0.17em}cos\hspace{0.17em}}\beta =\frac{A_y}{\left|\overline{A}\right|}\&\text{\hspace{0.17em}cos\hspace{0.17em}}\gamma =\frac{A_z}{\left|\overline{A}\right|}$

$A_x=4,A_y=5\&A_z=3$

$\therefore \overline{A}=A_x\widehat{i}+A_y\widehat{j}+A_z\widehat{k}\Rightarrow \overline{A}=4\widehat{i}+5\widehat{j}+3\widehat{k}.$