Answer
$\mathbf{r}\cdot\mathbf{F} = x\,y + y(-x) = 0.$
As a result, the arrows form a circular flow around the origin, and although the vectors
are not drawn to scale, their directions are in correct proportion, and do not intersect.
Work Step by Step
The given vector field is $\mathbf{F}(x,y)=y\,\mathbf{i}-x\,\mathbf{j}$.
At each point $(x,y)$, the corresponding vector is $\langle y,-x\rangle$.
This means that the vector at a point is obtained by rotating the position vector
$\mathbf{r}=\langle x,y\rangle$ by $90^\circ$ in the clockwise direction. Therefore,
each vector in the field is perpendicular to the position vector because
$\mathbf{r}\cdot\mathbf{F} = x\,y + y(-x) = 0.$
As a result, the arrows form a circular flow around the origin, and although the vectors
are not drawn to scale, their directions are in correct proportion, and do not intersect.
