Directional derivative for differentiable function

Question

In the directional derivative formula $$\frac{\partial f}{\partial v} = \nabla f \cdot v$$

why must $v$ be a unit vector?

Answer 1

In general, $v$ does not have to be a unit vector. But if $\|v\| = 1$, then $\|v\|$ doesn't artificially affects our geometric interpretations. In more general contexts (e.g., differential geometry), we define the gradient of $f$ in $p$ as the vector ${\rm grad}\, f(p)$ verifying $${\rm d}f_p(v) = \langle {\rm grad}\,f(p), v\rangle, \quad \forall v$$That is, we think "backwards". In $\Bbb R^n$, we have ${\rm grad}\,f(p) = \nabla f(p)$.