Directional Derivative(DD) If Iam going to derive the DD of a function without using a unit vector   i.e.$u=&lt;a,b&gt;$ (not unit vector). I get the same result $Du=(Fx).a+(Fy).b$.So why are we considering unit vector always?
If  V = &lt; a, b> is not a unit vector then afx + bfy is not the directional derivative of f at the direction of V. It is the directional derivative multiplied  by the norm of V. Therefore you do not get the same result unless you normalize V.
The formula stated here gives the derivative of f, a real valued function of two variables, with respect to a vector u whose components are 'a' and 'b' at a point where the partial derivatives(denoted in the question by fx and fy) of f are evaluated, in case the function f is differentiable there, as a function of two variables. If we put the additional condition that u is a unit vector we get what is usually referred to as the directional derivative of f,at the point, along u and which is equal to the component of the gradient vector of f along u.
Ref: Tom Apostol: Mathematical Analysis.