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

$x-\text{a}xis\to \cos \alpha =\frac{4}{5\sqrt{2}}$

$\cos \beta =\frac{{\text{a}}_y}{\left|a\right|}$                                       

$y-\text{a}xis\to \cos \beta =\frac{1}{\sqrt{2}}$

$z-\text{a}xis\to \cos \gamma =\frac{3}{5\sqrt{2}}$

$\cos \gamma =\frac{{\text{a}}_z}{\left|a\right|}$

$\overrightarrow{a}={\text{a}}_x\hat{i}+{\text{a}}_y\hat{j}+{\text{a}}_{\text{z}}\hat{k}$

$=\frac{4\hat{i}}{5\sqrt{2}}+\frac{\hat{j}}{\sqrt{2}}+\frac{3\hat{k}}{5\sqrt{2}}=\frac{\overline{a}}{\left|\overline{a}\right|}$

Giving vector

$=\frac{4\hat{i}+5\hat{j}+3\hat{k}}{5\sqrt{2}}=\frac{\overline{a}}{\left|\overline{a}\right|}$

$\text{a}=4\hat{i}+5\hat{j}+3\hat{k}$