Questions: Vector Fields and Their Representations

Question 1

What is the output of the vector field F(x, y) = ⟨−y, x⟩ at the point (3, 2)?

• A. A scalar value of −6
• B. The vector ⟨−2, 3⟩
• C. The vector ⟨3, 2⟩
• D. The vector ⟨−3, −2⟩

Answer: (B)

A vector field evaluates to a vector at each point, not a scalar. Substituting (x, y) = (3, 2) into F(x, y) = ⟨−y, x⟩ gives ⟨−2, 3⟩. The P-component is −y = −2, and the Q-component is x = 3. This illustrates that the output is a vector (two numbers), not a single number, and that the components of a vector field are functions of position.

Question 2

A fluid velocity field has arrows at every sampled point directed toward a single center point, with the arrows growing longer as they approach the center. What does this pattern most directly represent?

• A. A rotating flow circling the center point
• B. A uniform flow in the direction of the center
• C. A converging (sink) flow where fluid flows inward and accelerates toward the center
• D. A field with constant magnitude but varying direction

Answer: (C)

Arrows pointing toward a central point with increasing length indicate a sink: fluid is converging inward, and the flow speed increases as it approaches the center. This is the visual pattern of a draining vortex or gravitational attraction. Rotation would show arrows circling the center (tangent to circles), not pointing toward it. This example illustrates why arrow patterns in vector field visualizations carry geometric information about the field's behavior.

Question 3

In a vector field, two points that are very close together in space can have vectors pointing in completely different directions.

• T. True
• F. False

Answer: True

A vector field assigns an independent vector to each point in space based on the field's component functions P and Q (or P, Q, R in 3D). There is no requirement that nearby points have similar vectors — the field could change direction rapidly. In practice, smooth physical fields (like velocity or gravity) tend to vary continuously, so nearby points often have similar vectors, but this is a property of the specific field, not a definitional requirement. The field F(x, y) = ⟨sin(100x), cos(100y)⟩ would have rapidly oscillating directions at nearby points.

Question 4

The vector field F(x, y) = ⟨1, 0⟩ produces longer arrows at points farther from the origin, reflecting increasing field strength with distance.

• T. True
• F. False

Answer: False

F(x, y) = ⟨1, 0⟩ is a uniform field — every point in the plane gets the exact same vector ⟨1, 0⟩, regardless of location. The magnitude is 1 everywhere, and all arrows point in the positive x-direction with equal length. This is analogous to a uniform horizontal wind: same speed and direction throughout the region. Arrow length in a vector field diagram reflects the magnitude of F at that point, not distance from the origin.

Question 5

Why are vector fields more appropriate than individual vectors for modeling physical phenomena like gravity or fluid flow, and what does the function F: ℝⁿ → ℝⁿ structure capture?

Model answer: Gravity and fluid flow don't act at a single point — they assign a force or velocity to every point in space simultaneously. A single vector captures what happens at one location; a vector field captures the complete spatial structure by specifying a vector at every point. The function structure F: ℝⁿ → ℝⁿ formalizes this: you input a position and get the corresponding vector, making the whole spatial pattern computable and analyzable.

The power of the vector field concept is that it lifts the idea of a directed quantity from individual points to an entire region of space. This is what makes it possible to compute quantities like total work done by a force along a path (line integrals) or whether a fluid is spreading out or rotating at a given location (divergence and curl). Physical laws like Maxwell's equations and the Navier-Stokes equations are naturally expressed as relationships between vector fields, not individual vectors.