Projection of $\overrightarrow a$ on $\overrightarrow b$ is $\large\frac{\overrightarrow a.\overrightarrow b}{|\overrightarrow b|^2} $$\overrightarrow b $
If geometically interpreted, the scalar product of two vectors in the product of modulus of either vector and the projection of the other in its direction.
Thus, the projection of $\overrightarrow a$ on $\overrightarrow b$ is the dot product of $\overrightarrow a$ with the unit vector along $ \overrightarrow b$. is $\large\frac{\overrightarrow b}{|\overrightarrow b|}$
Therefore projection of $\overrightarrow a$ on $\overrightarrow b$ is $\large\frac{\overrightarrow a.\overrightarrow b}{|\overrightarrow b|^2}$$\:\hat b$
$=\large\frac{\overrightarrow a.\overrightarrow b}{|\overrightarrow b|^2}$$\:\overrightarrow b$
Hene the correct option is $A$