Why is this statement about vector fields correct?

Question

Let $\vec F$ be a continuous vector field in all $\mathbb{R}^2$, and let $C$ be a smooth curve, where in every point on it, the tangent to the curve $C$ is perpendicular to the field $\vec F$, then $\int_C\vec F\cdot dr=0$

I know that if $\vec F=Pi+Qj$ is a continuous vector field in a connected space, and it has a $U$ potential function such that $U_x=P,U_y=Q$. then $\vec F$ is conservative in that space.
How can I link what's being said to this theorem or another theorem about vector spaces?
What does it mean that the tangent to the curve is perpendicular to $\vec F$ in every point, what can I infer from that?
Would appreciate any help, Thanks in advance!

Answer 1

Hint:

1. When is $\vec u\cdot\vec v=0$, for two vectors $u,v\in\Bbb R^2$?
2. What is the direction of $d\vec{r}$?