I am confusing myself when it comes to directional derivatives and gradients. The gradient of a function shows the direction of the greatest change. So when we chose a unit vector as the direction to find the directional derivative, from my understanding, what we are doing is projecting the the gradient vector onto the directional vector. That projection is the directional derivative at that give point say $(a,b)$. Say also that the gradient is the vector g and that the vector u is the unit direction vector. Why is it that the gradient vector is projected on the unit vector thus giving us a the directional derivative in the direction of the unit vector u and not vice versa, that is the unit vector projects on the gradient giving us the directional derivative in that direction (of the gradient vector).
I have mixed everything up in my head and any help to clear things up is appreciated! Thanks :)