When working with directional derivatives, to obtain a unit vector, one divides the elements of the vector by its magnitude, giving the direction of the vector. However, what is the intuition behind this or perhaps what does it mean geometrically?
Thank you!
In general direction derivative with respect to a vector $\vec v$ is defined by
$$\frac{\partial f}{\partial \vec v}(\vec x_0)=\lim_{t\to 0}\frac{f(\vec x_0+t\vec v)-f(\vec x_0)}{t}$$
and by $\vec v = \lambda \hat u$ we obtain
$$\frac{\partial f}{\partial \vec v}(\vec x_0)=\frac{\partial f}{\partial \lambda \hat u}(\vec x_0)=\lambda \cdot\lim_{t\to 0}\frac{f(\vec x_0+t\lambda\hat u)-f(\vec x_0)}{\lambda t}=\lambda\cdot \frac{\partial f}{\partial \hat u}(\vec x_0)$$
therefore if we are interested to the directional derivative with reference to the unitary vector we need to divide by $\lambda=|\vec v|$.
When $f$ is differentiable
$$\frac{\partial f}{\partial \vec v}(\vec x_0)=\nabla f(\vec x_0)\cdot \vec v\implies \frac{\partial f}{\partial \lambda \hat u}(\vec x_0)=\lambda\nabla f(\vec x_0)\cdot \hat u$$