1. The problem is to understand and sketch the vector field given by $$\mathbf{F}(x,y) = \left\langle -\frac{y}{2}, \frac{x}{2} \right\rangle.$$\n\n2. This vector field assigns to each point $(x,y)$ a vector with components $-\frac{y}{2}$ in the $x$-direction and $\frac{x}{2}$ in the $y$-direction.\n\n3. To verify the vectors at given points, substitute the coordinates into the formula:\n- At $(2,0)$: $$\mathbf{F}(2,0) = \left\langle -\frac{0}{2}, \frac{2}{2} \right\rangle = \langle 0,1 \rangle.$$\n- At $(0,2)$: $$\mathbf{F}(0,2) = \left\langle -\frac{2}{2}, \frac{0}{2} \right\rangle = \langle -1,0 \rangle.$$\n- At $(-2,0)$: $$\mathbf{F}(-2,0) = \left\langle -\frac{0}{2}, \frac{-2}{2} \right\rangle = \langle 0,-1 \rangle.$$\n- At $(0,-2)$: $$\mathbf{F}(0,-2) = \left\langle -\frac{-2}{2}, \frac{0}{2} \right\rangle = \langle 1,0 \rangle.$$\n- At $(2,2)$: $$\mathbf{F}(2,2) = \left\langle -\frac{2}{2}, \frac{2}{2} \right\rangle = \langle -1,1 \rangle.$$\n- At $(-2,2)$: $$\mathbf{F}(-2,2) = \left\langle -\frac{2}{2}, \frac{-2}{2} \right\rangle = \langle -1,-1 \rangle.$$\n- At $(-2,-2)$: $$\mathbf{F}(-2,-2) = \left\langle -\frac{-2}{2}, \frac{-2}{2} \right\rangle = \langle 1,-1 \rangle.$$\n- At $(2,-2)$: $$\mathbf{F}(2,-2) = \left\langle -\frac{-2}{2}, \frac{2}{2} \right\rangle = \langle 1,1 \rangle.$$\n\n4. These match the vectors provided, confirming the correctness of the vector field formula and the vectors at the specified points.\n\n5. The vector field represents a rotation around the origin, with vectors perpendicular to the radius vector from the origin and magnitude proportional to the distance from the origin divided by 2.\n\nFinal answer: The given vectors correctly represent the vector field $$\mathbf{F}(x,y) = \left\langle -\frac{y}{2}, \frac{x}{2} \right\rangle.$$