Sorry for the trivial question. Say that I have two functions $g$ (differentiable) and $e$ from some interval $J$ to $\mathbb{R}^{3}$.
Is there any difference in either writing: $$ \forall t \in J \colon \; g'(t) \cdot (g(t) \times e(t)) = 0,$$ or, more succinctly: $$ g' \cdot (g \times e) = 0$$ ?
The second notation is widely accepted, and is usually assumed to mean exactly the first statement that you wrote. So yes, I would say that you are allowed to write $$g'\cdot(g\times e) = 0.$$