\[ \mathbf{r} = \mathbf{r} \left( t \right) \]
or in general if for each scalar variable there exist a vector value then
\[ \mathbf{A} = \mathbf{A} \left( t \right) \]
This relation can be inversed so that a scalar is a function of a vector. One example is temperature which can be described at every point so that
\[ T = T \left( \mathbf{r} \right) \]
or, again, in general if for each vector value there exist a scalar value
\[ \phi = \phi \left( \mathbf{r} \right) \]
In the case when \( \mathbf{r} \) is the position vector, then the scalar is a function of position.
A third example denotes a vector as a function of a vector. If one looks at the rigid body rotation that rotates at a constant angular velocity, \( \boldsymbol{\omega} \), then the velocity at a point on the body can be described as
\[ \mathbf{V} = \boldsymbol{\omega} \times \mathbf{r} \]
where the position vector, \( \mathbf{r} \), is taken from the axis of rotation.
Thus, the velocity vector, \( \mathbf{V} \), becomes a function of the position vector, \( \mathbf{r} \).
\[ \mathbf{V} = \mathbf{V} \left( \mathbf{r} \right) \]
Likewise, the general form becomes
\[ \mathbf{A} = \mathbf{A} \left( \mathbf{r} \right) \]
and when \( \mathbf{r} \) is the position vector \( \mathbf{A} \) is a vector function of position.
