📝 Vector Fields in Calculus (13 MCQs)

📖 From Calculus • 16. Topics in vector Calculus • 13 questions available

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What is Vector Fields in Calculus?

Vector Fields in Calculus:
Vector fields assign a vector to each point in space, often representing physical quantities like velocity or force, denoted as $\mathbf{F}(x,y) = P(x,y)\mathbf{i} + Q(x,y)\mathbf{j}$ in 2D or $\mathbf{F}(x,y,z) = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}$ in 3D.

Example:
The 2D vector field $\mathbf{F}(x,y) = -y\mathbf{i} + x\mathbf{j}$ represents counterclockwise rotation around the origin.

Reason:
This notation allows us to analyze physical phenomena like fluid flow or electromagnetic fields using calculus operations.

Easy

Medium

Hard

📝 All Vector Fields in Calculus MCQs

Q1. A vector field is defined by $\mathbf{F}(x,y)=\langle y,x \rangle$ . At the point $(2,-3)$ , which vector describes the field and its direction most accurately?

\langle -3,2\rangle

, pointing left and upward ✅

\langle 2,-3\rangle

, pointing right and downward

\langle -3,-2\rangle

, pointing left and downward

\langle 3,2\rangle

, pointing right and upward

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Substituting $x=2$ and $y=-3$ into $\mathbf{F}(x,y)=\langle y,x\rangle$ gives $\mathbf{F}(2,-3)=\langle -3,2\rangle$ . The negative first component means motion toward the left, while the positive second component means motion upward. Therefore option A correctly identifies both magnitude components and direction.

Q2. Consider $\mathbf{F}(x,y)=\langle x,-y\rangle$ . Which description best explains how the field behaves along the positive $x$ -axis and positive $y$ -axis?

A.Vectors point toward the origin on both axes

B.Vectors point away from the origin on both axes

C.Vectors point away from the origin on the

x

-axis and toward the origin on the

y

-axis ✅

D.Vectors point toward the origin on the

x

-axis and away from the origin on the

y

-axis

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: On the positive $x$ -axis, $y=0$ , so the field becomes $\langle x,0\rangle$ , pointing away from the origin. On the positive $y$ -axis, $x=0$ , giving $\langle 0,-y\rangle$ , which points downward toward the origin. Thus the two axes exhibit opposite behavior.

Q3. A velocity field for water flowing through a channel is $\mathbf{v}(x,y)=\langle 4-y,0\rangle$ . What does this model imply about the flow at points with increasing $y$ -coordinate?

A.Water speed increases as

y

increases

B.Water speed decreases as

y

increases while direction remains rightward ✅

C.Water reverses direction when

y>4

, but its speed never changes

D.Water moves vertically upward as

y

increases

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The horizontal component is $4-y$ , so the flow is entirely horizontal. For $y<4$ , the value is positive and the water moves rightward. As $y$ increases, the horizontal speed decreases. At $y=4$ , the velocity is zero, and for $y>4$ , the direction reverses. Hence B captures the model.

Q4. Suppose $\mathbf{F}(x,y)=\langle -y,x\rangle$ . A particle initially at $(1,0)$ moves according to the direction of the field. Which qualitative path behavior is most reasonable?

A.It initially moves upward and tends to circulate counterclockwise ✅

B.It initially moves downward and tends to circulate clockwise

C.It initially moves rightward and then stops

D.It moves directly toward the origin along a straight line

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: At $(1,0)$ , the vector is $\langle 0,1\rangle$ , so the particle initially moves upward. More generally, $\langle -y,x\rangle$ is perpendicular to the radial vector $\langle x,y\rangle$ , producing tangential motion around the origin. This is characteristic of counterclockwise circulation rather than radial motion.

Q5. For $\mathbf{F}(x,y)=\langle 2x,2y\rangle$ , a student claims that vectors become longer near the origin because both components increase there. Which evaluation is correct?

A.The claim is correct because both components approach zero

B.The claim is incorrect because the magnitude is

2\sqrt{x^2+y^2}

, so vectors become shorter near the origin ✅

C.The claim is correct only on the positive axes

D.The claim is incorrect because every vector has constant magnitude

2

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The magnitude of $\mathbf{F}(x,y)=\langle 2x,2y\rangle$ is $\sqrt{(2x)^2+(2y)^2}=2\sqrt{x^2+y^2}$ . As a point approaches the origin, both coordinates approach zero and therefore the vector magnitude also approaches zero. The student's reasoning reverses the actual relationship between position and vector length.

Q6. A field is given by $\mathbf{F}(x,y)=\langle x^2,y^2\rangle$ . At $(-2,1)$ , which conclusion follows from evaluating the field?

A.The vector points southwest

B.The vector points northwest

C.The vector points northeast ✅

D.The vector points southeast

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Evaluating gives $\mathbf{F}(-2,1)=\langle (-2)^2,1^2\rangle=\langle 4,1\rangle$ . Both components are positive, so the vector points to the right and upward, which is the northeast direction. The negative $x$ -coordinate of the point does not force the vector's first component to be negative because that component is squared.

Q7. A diagram of a vector field shows arrows that are horizontal everywhere, with arrows pointing right for $x<0$ , becoming zero at $x=0$ , and pointing left for $x>0$ . Which model best matches this pattern?

\mathbf{F}(x,y)=\langle x,0\rangle

\mathbf{F}(x,y)=\langle -x,0\rangle

✅

\mathbf{F}(x,y)=\langle y,0\rangle

\mathbf{F}(x,y)=\langle 1-x^2,0\rangle

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The arrows are horizontal, so the second component must be zero. For $x<0$ , the arrows point right, requiring a positive first component; for $x>0$ , they point left, requiring a negative first component. The expression $-x$ satisfies both conditions and becomes zero exactly at $x=0$ .

Q8. A plotted vector field has arrows tangent to circles centered at the origin. At the point $(0,3)$ , the arrow points left. Which field is most consistent with the plot?

\mathbf{F}(x,y)=\langle x,y\rangle

\mathbf{F}(x,y)=\langle y,-x\rangle

\mathbf{F}(x,y)=\langle -y,x\rangle

✅

\mathbf{F}(x,y)=\langle -x,-y\rangle

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: For $\mathbf{F}(x,y)=\langle -y,x\rangle$ , substituting $(0,3)$ gives $\langle -3,0\rangle$ , which points left. The field is tangent to circles because its vectors are perpendicular to the radial vector $\langle x,y\rangle$ . This matches both the displayed tangential pattern and the specified direction.

Q9. An engineer models wind by $\mathbf{W}(x,y)=\langle 6-y,x\rangle$ . At the location $(2,4)$ , a drone must initially travel with the wind. What direction should the drone expect?

A.Northeast ✅

B.Northwest

C.Southeast

D.Southwest

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: At $(2,4)$ , the wind vector is $\mathbf{W}(2,4)=\langle 6-4,2\rangle=\langle 2,2\rangle$ . Both components are positive, so the wind has equal eastward and northward components. Therefore its direction is northeast. The key is to evaluate the field first rather than infer direction only from the coordinates.

Q10. For $\mathbf{F}(x,y)=\langle -2y,2x\rangle$ , a student argues that the vectors point directly away from the origin because both components depend on the coordinates. How should this reasoning be corrected?

A.The vectors are radial because their magnitude is constant

B.The vectors are tangential because

\mathbf{F}\cdot\langle x,y\rangle=0

✅

C.The vectors are radial because both components are linear

D.The vectors point randomly because the components change sign

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: To determine whether a vector is radial or tangential, compare it with the radial vector $\langle x,y\rangle$ . Their dot product is $(-2y)x+(2x)y=0$ , so they are perpendicular wherever the field is defined. Therefore the vectors are tangent to circles centered at the origin, not directed radially.

Q11. Two vector fields are proposed for air moving around an obstacle: $\mathbf{F}_1=\langle -y,x\rangle$ and $\mathbf{F}_2=\langle x,y\rangle$ . Which comparison is most accurate for points away from the origin?

A.Both fields point radially outward

\mathbf{F}_1

is tangential to circles, while

\mathbf{F}_2

is radial ✅

\mathbf{F}_1

is radial inward, while

\mathbf{F}_2

is tangential

D.Both fields are tangential but rotate in opposite directions

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: For $\mathbf{F}_2=\langle x,y\rangle$ , the vector is parallel to the position vector, so it points radially outward. For $\mathbf{F}_1=\langle -y,x\rangle$ , the dot product with $\langle x,y\rangle$ is zero, making it perpendicular to the radial direction. Thus the first field is tangential while the second is radial.

Q12. A field is $\mathbf{F}(x,y)=\langle 3x,-3y\rangle$ . A particle moves along the line $y=x$ . At $x=2$ , what is the component of the field in the direction of the particle's path?

A.0 ✅

3\sqrt{2}

-3\sqrt{2}

D.-6

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Along $y=x$ , the path direction can be represented by the unit vector $\mathbf{u}=\frac{1}{\sqrt{2}}\langle 1,1\rangle$ . At $(2,2)$ , the field is $\langle 6,-6\rangle$ . Its directional component is $\mathbf{F}\cdot\mathbf{u}=\frac{6-6}{\sqrt{2}}=0$ . Thus the field has no component along that path.

Q13. A vector field is modeled as $\mathbf{F}(x,y)=\langle x(4-x),y(4-y)\rangle$ . A population movement model uses vector magnitude to represent movement intensity. At which point is the movement intensity zero?

A.Only at

(0,0)

B.At

(4,4)

only

C.At all points where

x\in{0,4}

and

y\in{0,4}

✅

D.At every point on the line

x=y

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The vector is zero only when both components vanish. The first component is zero when $x=0$ or $x=4$ , while the second is zero when $y=0$ or $y=4$ . Combining these conditions gives four points: $(0,0)$ , $(0,4)$ , $(4,0)$ , and $(4,4)$ . Therefore C is correct.

🔗 Related Topics (MCQs)

📝 Graphical Representations of Vector Fields MCQs 📝 Vector field notation and components MCQs 📝 Inverse square vector fields MCQs 📝 Gradient vector fields MCQs 📝 Conservative vector fields and potential functions MCQs

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