Does this definition of directional derivative depend on the magnitude?

Question

Here is the definition:

My calculus book defines directional derivatives for unit tangent vectors. According to Wikipedia, there is a convention that uses both the direction and magnitude. However, the definition says "in the direction $v$ at p". Does the magnitude somehow cancel out to have that $D_vf = D_uf$ for any tangent vectors $u$ and $v$ at the point $p \in \mathbb R^n$ that point in the same direction, or is magnitude really relevant here?

I think magnitude is relevant because I think $D_{[4,2]}f = 2D_{[2,1]}f$ and not $D_{[4,2]}f = D_{[2,1]}f$.

If magnitude is relevant, then should the definition instead be something like "with respect to the vector $v$ at p" ?

My book is An Introduction to Manifolds by Loring W. Tu.

Answer 1

The magnitude is relevant, as you thought. The terminology you suggested might be more clear.

Answer 2

We can write the non-zero vector $v = c \, u$ where $c \in \mathbb{R}^+$ and $u$ is the unit vector poinitng along the direction of $v$. Then $v = \langle v^i \rangle = c \, \langle u^i \rangle = c \, \langle 1, 1, 1 \rangle$.

Then $$ D_v f = \sum_{i} v^i \frac{\partial f}{\partial x^i} (p) = c \sum_{i} u^i \frac{\partial f}{\partial x^i} (p) = c \, D_u f. $$

So the magnitude is relevant in the definition.