LHS where the argument of the function isn't explicit stated (vector equation)

Question

The Lorentz force is given as $$ \mathbf F= q\left[\mathbf E(\mathbf r(t),t)+\mathbf v(t)\times \mathbf B(\mathbf r(t),t)\right] \tag 1 $$ where $\mathbf E, \mathbf B:\mathbb R^4\to\mathbb R^3$ are vector fields and $\mathbf r, \mathbf v:\mathbb R\to \mathbb R^3$ are vector-valued functions of one variable, $t\in \mathbb R$. And $q$ is a constant.

Question:

In books the argument of $\mathbf F$ is not explicit given, but why?

Does it mean it is a constant vector $\mathbf F\in \mathbb R^3$, i.e. $$ \mathbf F=(F_x,F_y,F_z) \quad ? \tag 2 $$

Or, based on the right hand side, is $\mathbf F$ a vector-valued function, $\mathbf F: \mathbb R\to \mathbb R^3$, i.e. $$ \mathbf F(t)=\big(F_x(t),F_y(t),F_z(t) \big) \quad ? \tag 3 $$

Or is it maybe a vector field, $\mathbf F: \mathbb R^4\to \mathbb R^3$, i.e. $$ \mathbf F(\mathbf r(t),t)=\big(F_x(r(t),t)),F_y(r(t),t)),F_z(r(t),t)) \big ) \quad ? \tag 4 $$

Answer 1

Your formula is a formula for the force acting on a particle given its path $r(t)$. Therefore, $F$ is a function of time, $F:t\to F(t)$. It is not a vector field or a constant.

Answer 2

It is a vector valued function, but it is defined on $\mathbb R^7$, $\mathbf F: \mathbb R^7\to\mathbb R^3$, $$\mathbf F=\mathbf F(t, \mathbf r,\mathbf v)$$