Can a unit vector be treated as a vector field?

Question

Say you have the kth component of a vector field:

$$\phi \hat{k}$$

Can we treat this as a scalar field $\phi$ multiplied by the vector field $\hat{k}$?

EDIT:

So, if you wanted to find the curl of $\phi \hat{k}$, could you use the following formula with $f$ = $\phi$ and $u$ = $\hat{k}$?

u is typically supposed to be a vector field, and f a scalar field.

Answer 1

Let's say your vector field $F:\mathbb{R}^3\to\mathbb{R}^3$ is given by $$ F(x,y,z)=f_1(x,y,z)i+f_2(x,y,z)j+f_3(x,y,z)k $$ where $f_j$ are scalar functions. Then yes, $G:\mathbb{R}^3\to\mathbb{R}^3$ with $$ G(x,y,z)=f_3(x,y,z)k $$ gives you another vector field. You could call the right-hand side "a scalar field multiplied by a constant vector". (You don't call $k$ a vector field here; it is nothing but a constant vector.)