📝 Vector field notation and components (13 MCQs)

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

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What is Vector field notation and components?

Vector field notation and components:
Vector fields use component functions $P, Q, R$ of variables $x, y, z$ , written as $\mathbf{F} = \langle P, Q, R \rangle$ , where each component is a scalar function.

Example:
For $\mathbf{F}(x,y) = \langle x^2, xy \rangle$ , $P = x^2$ and $Q = xy$ are the horizontal and vertical components.

Reason:
Component notation enables systematic application of calculus operations like divergence $\nabla \cdot \mathbf{F}$ and curl $\nabla \times \mathbf{F}$ .

Easy

Medium

Hard

📝 All Vector field notation and components MCQs

Q1. A wind model assigns the velocity $\mathbf{V}(x,y)=(3-y,2x)$ at every point. A researcher claims that the vector at $(2,1)$ is $(3,2)$ because the coefficients of $x$ and $y$ are 3 and 2. What is the correct interpretation?

A.The vector is

(2,1)

because the point coordinates are unchanged

B.The vector is

(2,4)

because each component must include both coordinates

C.The vector is

(2,4)

because

3-1=2

and

2(2)=4

✅

D.The vector is

(3,2)

because coefficients determine the vector directly

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: A vector field gives a vector after the coordinate values are substituted into its component functions. At $(2,1)$ , the first component is $3-1=2$ , while the second is $2(2)=4$ . Thus the field assigns $(2,4)$ to that location, not the coefficient pair $(3,2)$ .

Q2. Two vector fields are defined by $\mathbf{F}(x,y)=(x^2,y^2)$ and $\mathbf{G}(x,y)=(x^2-y^2,2xy)$ . At $(1,1)$ , which comparison is most accurate?

A.They produce identical vectors because both contain

x^2

and

y^2

\mathbf{F}

gives

(1,1)

, while

\mathbf{G}

gives

(0,2)

, so their local directions differ ✅

C.Both vectors have magnitude 2, so their directions must be identical

\mathbf{G}

gives

(1,1)

because

2xy=x^2+y^2

(1,1)

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Substituting $(1,1)$ into $\mathbf{F}$ gives $(1,1)$ , whereas substituting into $\mathbf{G}$ gives $(1-1,2)= (0,2)$ . This demonstrates why component formulas, rather than superficial similarities between expressions, determine the vector assigned at a specific point.

Q3. A navigation system uses $\mathbf{V}(x,y)=(2x-y,x+2y)$ . A drone moves from $P=(1,2)$ to $Q=(3,1)$ . Which statement correctly distinguishes displacement from the field vector at $P$ ?

A.Both are

(2,-1)

, because the field describes the drone's displacement

B.The displacement is

(2,-1)

, while

\mathbf{V}(1,2)=(0,5)

✅

C.The displacement is

(1,2)

, while

\mathbf{V}(1,2)=(2,-1)

D.The displacement is

(0,5)

, while

\mathbf{V}(1,2)=(2,-1)

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The displacement from $P$ to $Q$ is found by subtracting coordinates: $Q-P=(3-1,1-2)=(2,-1)$ . The field vector at $P$ is obtained separately: $\mathbf{V}(1,2)=(2(1)-2,1+2(2))=(0,5)$ . These represent different physical quantities.

Q4. A student rewrites $\mathbf{F}(x,y,z)=2x\mathbf{i}-3y\mathbf{j}+4z\mathbf{k}$ as the scalar expression $2x-3y+4z$ . Which error has occurred?

A.The student has changed a vector-valued field into a scalar-valued expression by removing the component directions ✅

B.The student has incorrectly squared all coordinates

C.The student has reversed the signs of all components

D.The student has changed the domain from three dimensions to two dimensions

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The symbols $\mathbf{i},\mathbf{j},\mathbf{k}$ identify independent coordinate directions. Removing them changes the mathematical object from a vector field to a scalar expression. Although the numerical component values remain $2x,-3y,4z$ , their directional information is essential to the vector field.

Q5. A force field is modeled by $\mathbf{F}(x,y)=(y,-x)$ . At the point $(2,-3)$ , an engineer wants to determine whether the force initially tends to move an object toward increasing or decreasing $x$ and $y$ . What conclusion follows?

A.The force is

(-3,-2)

, so both coordinates initially decrease ✅

B.The force is

(3,-2)

, so

x

increases while

y

decreases

C.The force is

(2,-3)

, so

x

increases while

y

decreases

D.The force is

(-3,2)

, so

x

decreases while

y

increases

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The field components are evaluated directly: the first component equals $y=-3$ , and the second equals $-x=-2$ . Hence $\mathbf{F}(2,-3)=(-3,-2)$ . Both components are negative, so the instantaneous tendency is toward decreasing $x$ and decreasing $y$ .

Q6. Consider a vector field $\mathbf{F}(x,y)=(x-y,x+y)$ . At which point does the field vector have zero first component but a positive second component?

(1,1)

✅

(-1,-1)

(1,-1)

(-1,1)

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: For the first component to vanish, $x-y=0$ , so $x=y$ . The second component must then be positive: $x+y=2x>0$ , requiring $x>0$ . Therefore $(1,1)$ satisfies both conditions, giving the vector $(0,2)$ .

Q7. A graph displays arrows of a two-dimensional field. Along the positive $x$ -axis, every arrow points directly upward, while along the positive $y$ -axis, every arrow points directly left. Which component pattern is most consistent with this observation?

A.The

x

-component is positive on both axes and the

y

-component is zero

B.The field could have the form

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

✅

C.The field must have the form

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

D.Both components must always be negative

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: For $\mathbf{F}(x,y)=(-y,x)$ , points on the positive $x$ -axis have $y=0$ , giving $(0,x)$ , which points upward. Points on the positive $y$ -axis have $x=0$ , giving $(-y,0)$ , which points left. This matches both graphical observations.

Q8. A temperature-dependent flow is modeled as $\mathbf{F}(x,y)=(x+2y,3x-y)$ . A student calculates $\mathbf{F}(-1,2)=(-5,-5)$ . Another calculates $(-3,-5)$ . Which evaluation is correct, and why?

A.The first is correct because

x+2y=-1+4=3

B.The second is correct because

x+2y=-1+4=3

and

3x-y=-5

✅

C.Neither is correct because the field cannot be evaluated at negative

x

D.The first is correct because both components must have the same sign

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Substitution gives the first component as $-1+2(2)=3$ , and the second as $3(-1)-2=-5$ . Therefore $\mathbf{F}(-1,2)=(3,-5)$ . The second student's stated vector is not correct either because its first component is $-3$ , despite using the correct arithmetic in the explanation.

Q9. Suppose a field is represented by $\mathbf{F}(x,y)=(2x,2y)$ . At two points $P=(1,2)$ and $Q=(2,4)$ , a student says the vectors differ because the points differ. Which stronger conclusion is justified?

A.The vectors are

(2,4)

and

(4,8)

, so they have the same direction but different magnitudes ✅

B.The vectors are identical because both points lie on the same line

C.The vectors have different directions because both coordinates change

D.The vectors are opposites because

Q=2P

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: At $P$ , the field is $(2,4)$ ; at $Q$ , it is $(4,8)=2(2,4)$ . Therefore the vectors have different magnitudes but the same direction. This requires recognizing how scaling the input point affects a field whose components are proportional to the coordinates.

Q10. A model uses $\mathbf{F}(x,y,z)=(yz,xz,xy)$ . At $P=(1,-2,3)$ , an analyst claims the second component is $-6$ because $xz=(-2)(3)$ . What is the correct second component?

A.-6

B.-3

C.3 ✅

D.-1

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The second component is $xz$ . At $P=(1,-2,3)$ , the coordinates required are $x=1$ and $z=3$ , so $xz=1(3)=3$ . The incorrect calculation uses $y$ in place of $x$ , a common component-indexing error.

Q11. A field is written as $\mathbf{F}=F_x\mathbf{i}+F_y\mathbf{j}$ . At a point, $F_x<0$ and $F_y>0$ . A student concludes that the vector points southwest because the first component dominates. What can be concluded without knowing their magnitudes?

A.It must point exactly west

B.It must point exactly north

C.It must lie in the northwest quadrant ✅

D.It must point southwest

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: A negative $x$ -component means motion toward the left, while a positive $y$ -component means motion upward. Therefore the vector lies in the northwest quadrant. The relative sizes determine its exact angle, but they cannot change the quadrant as long as neither component is zero.

Q12. Two methods are proposed for evaluating $\mathbf{F}(x,y)=(x^2-y,2xy)$ at $(2,3)$ . Method I substitutes coordinates into each component separately. Method II first computes $x^2-y+2xy$ and treats the result as one vector component. Which assessment is correct?

A.Both methods are equivalent because a vector has only one numerical value

B.Method I is valid; Method II incorrectly combines distinct vector components into one scalar ✅

C.Method II is valid because all components should be added

D.Neither method works unless the field is constant

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: A vector field contains separate directional components. Method I correctly produces $(2^2-3,2(2)(3))=(1,12)$ . Method II combines these directional quantities into a single scalar, losing the distinction between the $x$ - and $y$ -directions and therefore changing the mathematical meaning.

Q13. A field satisfies $\mathbf{F}(x,y)=(a x-b y,b x+a y)$ , where $a$ and $b$ are constants. At every point, the field is observed to be perpendicular to the position vector $(x,y)$ . Which condition on $a$ and $b$ must hold for all nonzero points?

a=0

✅

b=0

a=b

a+b=0

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Perpendicularity requires the dot product $(x,y)\cdot(ax-by,bx+ay)$ to vanish for every $x,y$ . Expanding gives $ax^2-bxy+bxy+ay^2=a(x^2+y^2)$ . For every nonzero point, this is zero only when $a=0$ . Then the field becomes $(-by,bx)$ , which is always perpendicular to the position vector.

🔗 Related Topics (MCQs)

📝 Vector Fields in Calculus MCQs 📝 Graphical Representations of Vector Fields MCQs 📝 Inverse square vector fields MCQs 📝 Gradient vector fields MCQs 📝 Conservative vector fields and potential functions MCQs

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