1. **(Section 17.1)** Match the vector fields with their sketches below by placing the letter of the function in the corresponding blank:\[\begin{array}{lll}\text{(I)} & \vec{F}(x, y) = \langle -y, -x \rangle & \\\text{(II)} & \vec{F}(x, y) = \langle y^2, x \rangle & \\\text{(III)} & \vec{F}(x, y) = \langle y, -x \rangle & \\\text{(IV)} & \vec{F}(x, y) = \langle x + y, y - x \rangle & \\\end{array}\]- **Vector Field (A):** - **Vector Field (B):**- **Vector Field (C):**- **Vector Field (D):**### Explanation of Vector Field Diagrams**Vector Field (A):** - The arrows form concentric circular patterns around the origin, indicating a spiral or rotational field.**Vector Field (B):** - The arrows are directed outward from the center, suggesting a divergence from the origin.**Vector Field (C):** - The vectors appear to create a clockwise circular motion around the origin, potentially indicative of a curl or vortex.**Vector Field (D):** - The pattern shows arrows converging towards the center, suggesting an inward flow to the origin.

We need to match the given vector field with their corresponding sketches. Any point on the Cartesian coordinate system can be represented as: x, y. The vector: x, y is obtained by joining the origin0,0 and the point: x, y. The tail of the vector is at the origin and the head of the vector is at x, y. It implies that the vector is directed away from the origin to the point. The vector field is a cluster of arrows, each with a specific magnitude and direction, that are associated with the points in the plane.Follow the steps below to obtain the sketch of a vector field. Draw x, y axes and set the scale of the graph such that -2≤x≤2 and -2≤y≤2. Take arbitrary values for x, y and calculate the corresponding vector at that point using the equation of F→(x, y). Draw a small arrow at that point as per the vector. Repeat this for several values of x, y. Plot the vector fields one after the other to obtain as follows.Use the steps above to make the vector field of F→(x, y)=-y, -x as follows. It matches with option B.Use the steps above to make the vector field of F→(x, y)=y2, x as follows. It matches with option C.Use the steps above to make the vector field of F→(x, y)=y, -x as follows. It matches with option A.Use the steps above to make the vector field of F→(x, y)=x+y, y-x as follows. It matches with option D. Hence, the matching can be done as follows. Vector field Sketch I B II C III A IV D Use this table to fill in the blanks.