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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. $b_x,b_y,b_z:\mathbb R \rightarrow \mathbb R$ .

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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.

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