Cross product of vector functions of different (input) dimensions?

Question

If I take the cross product of a vector function $\mathbf r(t)$ and a vector field $\mathbf A(x,y,z,t)$, is the result itself a vector field?

What is in general true if we take the cross product of vector functions/vector fields with different dimensions? What is the dimension of the cross product?

I.e. the cross product of ($n\neq p$) $$ \mathbf B:\mathbb R^n\rightarrow \mathbb R^3 \text{ and }\mathbf C:\mathbb R^p\rightarrow \mathbb R^3 $$ Or the cross product of ($n=p$) $$ \mathbf D:\mathbb R^n\rightarrow \mathbb R^3 \text{ and }\mathbf E:\mathbb R^n\rightarrow \mathbb R^3 $$

Answer 1

Since $\mathbf B$ and $\mathbf C$ are vectors in $\mathbb{R}^3$ the cross product $\mathbf B\times \mathbf C$ is a vector that depends from the two vector field and is a vector field $\mathbf B\times \mathbf C:\mathbb{R}^n \times\mathbb{R}^p \rightarrow \mathbb{R}^3$ .

The case of $\mathbf D$ and $\mathbf E$ is a special case.