I can't understand when an assigned triplet of scalar fields $$[a_x \: a_y \: a_z](x,y,z)$$ can represent a vector field $$\overline{a(x,y,z)}$$ There are some requirements for the triplet of scalar fields, in order for them to be the components of a vector field?
Edit 1: What my professor said
A triplet of scalar fields $\textbf{a}=(a_x\: a_y \: a_z)$ represents the scalar components of a vectorial field only if, given the versor $$\hat{\textbf{n}}=n_x\hat{\textbf{i}}+n_y\hat{\textbf{j}}+n_z\hat{\textbf{k}}$$ the scalar component of $\textbf{a}$ in the direction of $\hat{\textbf{n}}$ is $$a_n=n_xa_x+n_ya_y+n_za_z$$
Thanks in advance for the answers!