If I take the cross product of a constant vector and a vector field, is the result itself a constant vector or a vector field?
I.e. say I have a constant vector $\mathbf a=(x,y,z)\in \mathbb R^3$ and a vector field $\mathbf B:\mathbb R^3\rightarrow\mathbb R^3$.
The cross product is explicit \begin{align} \mathbf a\times \mathbf B(x,y,z)= &\hat e_x(yB_3(x,y,z)-zB_2(x,y,z))\\ &-\hat e_y(xB_3(x,y,z)-zB_1(x,y,z))\\ &\hat e_z(xB_2(x,y,z)-yB_1(x,y,z)) \end{align}
Is this result a new constant vector, let call it $\mathbf c$, $\mathbf c=(x,y,z)\in\mathbb R^3$?
Or is it a new vector field, say, $\mathbf C$, so $\mathbf C:\mathbb R^3\rightarrow \mathbb R^3?$?
Let $\vec{a}=(a_1,a_2,a_3)$, where $a_i$ are constant, and $ \vec{B}:\mathbb{R}^3\rightarrow\mathbb{R}^3 $, where $\vec{B}(x,y,z) = (B_1(x,y,z), B_2(x,y,z),B_3(x,y,z))$. Then, the cross product: $$ \vec{a}\times\vec{B}(x,y,z) = \hat{e}_x[a_2 B_3(x,y,z)-a_3B_2(x,y,z)] - \hat{e}_y[a_1B_3(x,y,z)-a_3B_1(x,y,z)] + \hat{e}_z[a_1B_2(x,y,z)-a_2B_1(x,y,z)] $$ where $(\hat{e}_x,\hat{e}_y,\hat{e}_z)=(\hat{i},\hat{j},\hat{k})$ are constants. This makes it a little clearer. So, we can rewrite this as a vector field $v:\mathbb{R}^3\rightarrow\mathbb{R}^3$, where $$ v(x,y,z):=\vec{a}\times\vec{B}(x,y,z) $$ Thus, for example, if $p=(p_1,p_2,p_3)\in\mathbb{R}^3$ is a fixed vector, then $$ v(p_1,p_2,p_3)=\vec{a}\times\vec{B}(p_1,p_2,p_3)$$
So indeed it is a vector field.