# Difference between $\mathbb R$ and $\mathbb R \rightarrow \mathbb R^3$ for vectors?

I wondering if I have understand the following notation correctly:

With a vector $\mathbf a\in \mathbb R^3$ we mean a constant vector: $$\mathbf a= a_x \hat e_x+ a_y \hat e_y + \hat e_z=(a_x,a_y,a_z),$$ where $a_x,a_y,a_z$ are constants (numbers/scalars), i.e. $a_x,a_y,a_z\in \mathbb R$.

With a vector $\mathbf b: \mathbb R \rightarrow \mathbb R^3$, we mean a vector function: $$\mathbf b(t) = b_x(t) \hat e_x + b_y(t) \hat e_y + b_z(t) \hat e_z=(b_x(t),b_y(t),b_z(t)),$$ where $b_x(t), b_y(t), b_z(t)$ are functions of one variable, i.e. $b_x,b_y,b_z:\mathbb R \rightarrow \mathbb R$ .

Your ${\bf b}$ is not a vector in ${\mathbb R}^3$ but a vector-valued function. As such it could be an element of a vector space of such functions, hence a vector of some other type; but this is not what your professor had in mind.