If the flow lines for the vector field $\vec { F } ( \vec { r } )$ are all concentric circles centered at the origin, then $\vec { H } ( \vec { r } ) \cdot \vec { r } = 0$ for all $\vec { r }$

Question

Quizplus · Accepted Answer

If the flow lines of $\vec{F}(\vec{r})$ are concentric circles centered at the origin, then $\vec{F}(\vec{r})$ is always tangent to these circles. Since $\vec{r}$ points radially outward from the origin, it is perpendicular to the tangent vector $\vec{F}(\vec{r})$ at every point on a circle. Therefore, their dot product $\vec{F}(\vec{r}) \cdot \vec{r} = 0$, indicating orthogonality.

If the Flow Lines for the Vector Field F⃗(r⃗)\vec { F } ( \vec { r } )F(r)

If the Flow Lines for the Vector Field $\vec { F } ( \vec { r } )$