Can a vector and its curl be collinear?

Question

While I was studying fluid mechanics and doing some vector calculus. I wondered if the following statement is true or false.

Given that $A$ is a smooth vector field and given that $V\times ( \nabla \times V)=0$. We must have $\nabla \times V=0$

Answer 1

Tip: Consider the field $(z,x,y)$ at point $(1,1,1)$.