How to define a vector operator?

Question

I know how to define a single-variable differential operator like

Clear[Op]  
Op[x_] := a D[#, {x, 1}] &

and I can do operations like

Op[x][f[x]]         (* af'[x]    *)
Op[x][Op[x][f[x]]]  (* a^2f''[x] *)

However, how about if I want to do a vectorial operator, i.e. something like

$$ \hat O = \vec a \cdot \vec \nabla_x, $$

so that when operating on some scalar function of $x = |\vec x|$, e.g. $f(x)$ we have

$$ \hat O f(x) = \vec a \cdot \vec \nabla_x f(x) = f'(x) \frac{\vec a \cdot \vec x}{x}\, ,\\ \left(\hat O\right)^2 f(x) = \left(\vec a \cdot \vec \nabla_x\right)^2 f(x) = \left( f''(x) - \frac{f'(x)}{x} \right) \frac{(\vec a \cdot \vec x)^2}{x^2} + \frac{f'(x)}{x} a^2\, . $$

How do I define an operator like this?

I would prefer operations to be done symbolically, without the need to specify a coordinate system.