The term direction vector seems to have ambiguous meaning or at least to be used in ambiguous ways (in my native language). A direction vector seems to be two different things in different contexts. Two examples:
Example 1: A force has a direction vector. It is the direction of the force, let's call it $\vec T$, so that the parametric destription of that force $\vec F$ is:$$\vec F=\vec{OP}+t\cdot\vec T\,,\qquad t\in \mathbb{R},$$where $\vec{OP}$ is a vector from the coordinate-system origin to the point of application of that force.
Example 2: A position in space can be defined as a vector $\vec s$ from the coordinate-system origin to the point. The direction vector of position is the direction in which it changes, so essentially $$\vec s\,'=\frac{\mathrm d\vec s}{\mathrm dt} .$$
If both of these uses are valid simultaneously, then $d\vec F/dt=\vec T$. But according to the first example, this only holds true for constant $\vec T$, which isn't the case for a varying vector as in example 2.
As explained, I believe I have seen both versions of the term direction vector used. Both as the direction of the vector as well as the direction which the vector changes towards. Possibly, I have misunderstood (or terms have been misused). I'd like to have cleared out exactly how this term is defined and used in mathematics in English.
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$\begingroup$Given a fixed nonzero vector $\mathbf v$, "the direction vector of/corresponding to $\mathbf v$" would typically refer to the unit (length/magnitude) vector in the same direction: $\dfrac{\mathbf v}{\Vert\mathbf v\Vert}$.
But in many other contexts, depending on the source, the phrase "direction vector" might be used without the requirement of being a unit vector.
For example, a discussion of the parametric vector form of a line, such as $\mathbf a+t\mathbf v$ might call $\mathbf v$ (in context) "a/the direction vector of/for the line". Relatedly, if $\mathbf s(t)$ is a vector function, then I wouldn't be surprised if a source said something like "the direction vector for $s$ at time $t_0$" to refer to $\mathbf s'(t_0)$, which is a direction vector for the line tangent to the parametric curve at $\mathbf s(t_0)$. If we wanted to normalize it, it's common to refer to the tangent unit vector (or unit tangent vector) $\mathbf T(t_0)=\dfrac{\mathbf s'(t_0)}{\Vert \mathbf s'(t_0)\Vert}$.
One place of caution, because textbooks differ, is in the definition of the directional derivative of a function of multiple variables. It may or may not expect you to normalize the vector for the direction first (or may simply not be defined for non-unit direction vectors).
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