Describing a Vector Field in the Plane
Describe the vector field \(\mathbf{F}=\mathbf{F}(x,y)=-y\mathbf{i}+x\mathbf{j}\) by drawing some of the vectors \(\bf{F}.\)
| \((x,y)\) | \(( 2,0)\) | \(( 0,2)\) | \(( -2,0)\) | \(( 0,-2)\) | \(( 3,3)\) | \(( -3,3)\) | \(( -3,-3)\) | \(( 3,-3)\) |
|---|---|---|---|---|---|---|---|---|
| \(\mathbf{F}(x,y) \) | \(\ 2\mathbf{j}\) | \(-2\mathbf{i}\) | \(-2\mathbf{j}\) | \(2\mathbf{i}\) | \(-3\mathbf{i}+3\mathbf{j}\) | \(-3\mathbf{i}-3\mathbf{j}\) | \(3\mathbf{i}-3\mathbf{j}\) | \(3\mathbf{i}+3\mathbf{j}\) |
Figure 3 illustrates the vectors from the table. Notice that each vector in the field is tangent to a circle centered at the origin, and the direction of the vectors indicates that the field is rotating counterclockwise. This field might represent the motion of a wheel spinning on an axle, with each vector equal to the velocity at a point of the wheel.