91Ó°ÊÓ

The vector field \(\mathbf{F}\) is given as \(\mathbf{F}=\langle x+y, y\rangle\). This means that the \(x\) component of the field is \(x+y\), and the \(y\) component is \(y\). We'll use these components to calculate the vectors at different points of the cartesian plane.

Let's choose some points in the cartesian plane to evaluate the components of the vector field: 1. Origin \((0,0)\) 2. \((1,0)\) 3. \((0,1)\) 4. \((1,1)\) 5. \((-1,0)\) 6. \((0,-1)\) 7. \((-1,-1)\) 8. \((-1,1)\) 9. \((1,-1)\)

Now, let's calculate the \(x\) and \(y\) components of the vector field for each of the selected points: 1. At \((0,0)\): \(\mathbf{F}(0,0) = \langle 0+0, 0\rangle = \langle 0,0\rangle\) 2. At \((1,0)\): \(\mathbf{F}(1,0) = \langle 1+0, 0\rangle = \langle 1,0\rangle\) 3. At \((0,1)\): \(\mathbf{F}(0,1) = \langle 0+1, 1\rangle = \langle 1,1\rangle\) 4. At \((1,1)\): \(\mathbf{F}(1,1) = \langle 1+1, 1\rangle = \langle 2,1\rangle\) 5. At \((-1,0)\): \(\mathbf{F}(-1,0) = \langle -1+0, 0\rangle = \langle -1,0\rangle\) 6. At \((0,-1)\): \(\mathbf{F}(0,-1) = \langle 0-1, -1\rangle = \langle -1,-1\rangle\) 7. At \((-1,-1)\): \(\mathbf{F}(-1,-1) = \langle -1-1, -1\rangle = \langle -2,-1\rangle\) 8. At \((-1,1)\): \(\mathbf{F}(-1,1) = \langle -1+1, 1\rangle = \langle 0,1\rangle\) 9. At \((1,-1)\): \(\mathbf{F}(1,-1) = \langle 1-1, -1\rangle = \langle 0,-1\rangle\)

Now we'll sketch the vector field by drawing arrows at each of the chosen points. The tail of each arrow coincides with the corresponding point, and the tip of each arrow represents the calculated vector at that point. When drawing arrows keep in mind the proportional length and direction as per the components. Once you have plotted arrows at the selected points, you should get the general idea of the vector field's behavior and the sketch should represent the given vector field \(\mathbf{F}=\langle x+y, y\rangle\).