Which of the following best describes the direction of the vector field $f(x,y)$ = $\frac{yi-xj}{x^2+y^2}$ at any point (x, y) in the plane?

• A. The vectors are always parallel to the $x$-axis.
• B. The vectors always point directly toward the origin.
• C. The vectors always point directly away from the origin.
• D. The vectors are always tangent to circles centered at the origin.

Answer: (D)

Verified step by step guidance

1. Identify the given vector field: \(\mathbf{f}(x,y) = \frac{y \mathbf{i} - x \mathbf{j}}{x^{2} + y^{2}}\).
2. Recall that the denominator \(x^{2} + y^{2}\) represents the square of the distance from the origin to the point \((x,y)\), so the vector field depends on the position relative to the origin.
3. Analyze the direction of the vector at any point \((x,y)\): the vector components are \((y, -x)\), which is perpendicular to the radius vector \((x, y)\) because their dot product is zero: \(x \cdot y + y \cdot (-x) = xy - xy = 0\).
4. Since the vector is perpendicular to the radius vector, it must be tangent to the circle centered at the origin passing through \((x,y)\).
5. Therefore, the vector field at any point is tangent to circles centered at the origin, meaning the vectors form a rotational pattern around the origin.