Precalculus Quiz: Finding Components Of Vectors

20 questions · exam conditions

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Finding Components Of VectorsQuestion 1 of 20

A robot moves along a path defined by vectors. It starts at $S(1, 4)$ and moves to $T(7, 1)$ , then to $U(-2, 8)$ . What are the components of the vector that would take the robot directly from its starting position to its final position?

(5, -3)

(-3, 4)

(6, -3)

(-9, 7)

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Precalculus Quiz

Precalculus Quiz: Finding Components Of Vectors

Practice Finding Components Of Vectors in Precalculus with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Finding Components Of Vectors, giving you a quick way to practice the rules, question types, and explanations that matter most for Precalculus.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

$(5, -3)$
$(-3, 4)$ (correct answer)
$(6, -3)$
$(-9, 7)$

Explanation: When you see a vector problem involving multiple movements, focus on the net displacement—the single vector that connects the starting point directly to the final destination, regardless of the path taken. The robot starts at

S(1, 4)

and ends at

U(-2, 8)

after passing through point

T

. To find the displacement vector from start to finish, you subtract the starting coordinates from the ending coordinates:

\overrightarrow{SU} = U - S = (-2, 8) - (1, 4) = (-3, 4)

. This gives you the components of the vector that would take the robot directly from

S

U

. Looking at the wrong answers: Choice A,

(5, -3)

, appears to be the vector from

S

T

with a sign error. Choice C,

(6, -3)

, is actually the vector

\overrightarrow{ST} = (7, 1) - (1, 4) = (6, -3)

—this only gets you to the intermediate point

T

, not the final destination. Choice D,

(-9, 7)

, is the vector

\overrightarrow{TU} = (-2, 8) - (7, 1) = (-9, 7)

, which represents the second leg of the journey from

T

U

, not the complete displacement. The correct answer is B,

(-3, 4)

. Study tip: For displacement problems, always work with initial and final positions only. The intermediate points are irrelevant for finding the net displacement vector. Remember: displacement equals final position minus initial position.

Question 2

Point $A$ is located at $(0, 3)$ and point $B$ is located at $(5, 3)$ . For the vector described, what is the component form of $\overrightarrow{AB}$ ?

$\langle 5, 0\rangle$ (correct answer)
$\langle -5, 0\rangle$
$\langle 0, 5\rangle$
$\langle 5, 3\rangle$

Explanation: This question tests understanding of how to find vector components by subtracting the coordinates of the initial point from the coordinates of the terminal point. A directed line segment from point A to point B represents a vector whose components are the differences in coordinates: ⟨final x - initial x, final y - initial y⟩. The horizontal component 5 - 0 = 5 tells us the vector moves 5 units right, and the vertical component 3 - 3 = 0 tells us it moves 0 units up or down. Choice A is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨5 - 0, 3 - 3⟩ = ⟨5, 0⟩. Choice B reverses the subtraction order, calculating ⟨x₁ - x₂, y₁ - y₂⟩, which gives the opposite vector (same magnitude, opposite direction). Key to finding vector components: always subtract initial point from terminal point (terminal - initial), and remember that the first component is the change in x, the second is the change in y. Check your work by visualizing: if point B is to the right of point A, the horizontal component should be positive; if B is above A, the vertical component should be positive.

Question 3

A vector has initial point $A(5, -1)$ and terminal point $B(0, 3)$ . For the vector described, what is the component form of $\overrightarrow{AB}$ ?

$\langle 5, -4 \rangle$
$\langle 0, 3 \rangle$
$\langle -5, 4 \rangle$ (correct answer)
$\langle -4, 5 \rangle$

Explanation: This question tests understanding of how to find vector components by subtracting the coordinates of the initial point from the coordinates of the terminal point. To find vector components, calculate the change in x (horizontal displacement) and the change in y (vertical displacement) by subtracting initial from terminal: horizontal component = x₂ - x₁, vertical component = y₂ - y₁. Using the formula for components, we substitute the given coordinates: ⟨0 - 5, 3 - (-1)⟩ = ⟨-5, 4⟩. Choice C is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨0 - 5, 3 - (-1)⟩ = ⟨-5, 4⟩. Choice A reverses the subtraction order, calculating ⟨5 - 0, -1 - 3⟩, which gives the opposite vector (same magnitude, opposite direction). Remember that vector components are not the same as point coordinates—components represent displacement (change in position), while coordinates represent location. The negative sign in a component is meaningful: negative horizontal component means leftward motion, negative vertical component means downward motion.

Question 4

A surveyor measures the position of three landmarks. Landmark A is at $(0, 0)$ , landmark B is at $(8, 6)$ , and landmark C is at $(5, -2)$ . If a fourth landmark D is positioned such that $\vec{AC} = 2\vec{AD}$ , what are the coordinates of landmark D?

$(5, -2)$
$(10, -4)$
$(2.5, -1)$ (correct answer)
$(-2.5, 1)$

Explanation: When you encounter vector equations with landmarks or points, you're working with vector algebra where vectors represent displacements between points. The key insight is that

\vec{AC} = 2\vec{AD}

means vector

\vec{AC}

is twice as long as vector

\vec{AD}

in the same direction. First, find vector

\vec{AC}

. Since A is at

(0,0)

and C is at

(5,-2)

, we have

\vec{AC} = (5-0, -2-0) = (5, -2)

. Now use the given relationship

\vec{AC} = 2\vec{AD}

. This means

\vec{AD} = \frac{1}{2}\vec{AC} = \frac{1}{2}(5, -2) = (2.5, -1)

. Since

\vec{AD}

represents the displacement from A to D, and A is at the origin, the coordinates of D are simply

(2.5, -1)

. Looking at the wrong answers: A)

(5, -2)

gives you the coordinates of point C, which happens when students confuse

\vec{AC}

with the position of D. B)

(10, -4)

results from incorrectly thinking

\vec{AD} = 2\vec{AC}

instead of the given relationship. D)

(-2.5, 1)

comes from incorrectly applying the negative of the correct vector, possibly from confusion about vector direction. The correct answer is C)

(2.5, -1)

. Strategy tip: When working with vector equations involving positions, always distinguish between vectors (displacements) and points (coordinates). Write out the vector components explicitly before applying any given relationships, and remember that if a vector starts at the origin, its components directly give the endpoint coordinates.

Question 5

Based on the coordinates, if the initial point is $A(-4, 0)$ and the terminal point is $B(3, 5)$ , what are the vector components of the vector from $A$ to $B$ ?

$\langle 7, 5 \rangle$ (correct answer)
$\langle -7, -5 \rangle$
$\langle -1, 5 \rangle$
$\langle 3, 5 \rangle$

Explanation: This question tests understanding of how to find vector components by subtracting the coordinates of the initial point from the coordinates of the terminal point. A directed line segment from point A to point B represents a vector whose components are the differences in coordinates: ⟨final x - initial x, final y - initial y⟩. The horizontal component 3 - (-4) = 7 tells us the vector moves 7 units right, and the vertical component 5 - 0 = 5 tells us it moves 5 units up. Choice A is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨3 - (-4), 5 - 0⟩ = ⟨7, 5⟩. Choice B reverses the subtraction order, calculating ⟨x₁ - x₂, y₁ - y₂⟩, which gives the opposite vector (same magnitude, opposite direction). Key to finding vector components: always subtract initial point from terminal point (terminal - initial), and remember that the first component is the change in x, the second is the change in y. To avoid errors, clearly label which point is initial and which is terminal, then methodically compute x₂ - x₁ for the horizontal component and y₂ - y₁ for the vertical component.

Question 6

A vector $\vec{v}$ has initial point $P(2, -3)$ and terminal point $Q(7, 1)$ . If this vector is translated so that its initial point becomes $R(-1, 4)$ , what are the coordinates of the new terminal point?

$(4, 8)$ (correct answer)
$(6, 0)$
$(-6, 0)$
$(8, -7)$

Explanation: First, find the components of vector

\vec{v}

\vec{v} = (7-2, 1-(-3)) = (5, 4)

. When a vector is translated, its components remain the same. If the new initial point is

R(-1, 4)

and the vector components are

(5, 4)

, then the new terminal point is

(-1+5, 4+4) = (4, 8)

. Choice B incorrectly uses the original terminal point coordinates. Choice C represents the negative of the correct displacement. Choice D results from subtracting instead of adding the components.

Question 7

Point $A$ is located at $(1, -5)$ and point $B$ is located at $(4, -1)$ . For the vector described, what is the component form of $\overrightarrow{AB}$ ?

$\langle 3, 4 \rangle$ (correct answer)
$\langle -3, -4 \rangle$
$\langle 5, -6 \rangle$
$\langle 4, -1 \rangle$

Explanation: This question tests understanding of how to find vector components by subtracting the coordinates of the initial point from the coordinates of the terminal point. To find vector components, calculate the change in x (horizontal displacement) and the change in y (vertical displacement) by subtracting initial from terminal: horizontal component = x₂ - x₁, vertical component = y₂ - y₁. The horizontal component 4 - 1 = 3 tells us the vector moves 3 units right, and the vertical component -1 - (-5) = 4 tells us it moves 4 units up. Choice A is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨4 - 1, -1 - (-5)⟩ = ⟨3, 4⟩. Choice B reverses the subtraction order, calculating ⟨1 - 4, -5 - (-1)⟩, which gives the opposite vector (same magnitude, opposite direction). Remember that vector components are not the same as point coordinates—components represent displacement (change in position), while coordinates represent location. The negative sign in a component is meaningful: negative horizontal component means leftward motion, negative vertical component means downward motion.

Question 8

Points $P$ , $Q$ , and $R$ are collinear with $Q$ between $P$ and $R$ . If $P(-1, 2)$ , $Q(3, 5)$ , and the vector $\vec{PQ}$ has the same direction as vector $\vec{QR}$ , what could be the coordinates of point $R$ if $|\vec{QR}| = 2|\vec{PQ}|$ ?

$(7, 8)$
$(11, 11)$ (correct answer)
$(5, 7)$
$(-1, 2)$

Explanation: This question tests vector operations and collinear points. When points are collinear with the same direction, you're essentially extending a line segment by a specific scale factor. First, find vector

\vec{PQ}

: From

P(-1, 2)

Q(3, 5)

, we get

\vec{PQ} = (3-(-1), 5-2) = (4, 3)

. The magnitude is

|\vec{PQ}| = \sqrt{4^2 + 3^2} = 5

. Since

\vec{QR}

has the same direction as

\vec{PQ}

and

|\vec{QR}| = 2|\vec{PQ}|

, we have

\vec{QR} = 2\vec{PQ} = 2(4, 3) = (8, 6)

. To find point

R

, add this vector to

Q

R = Q + \vec{QR} = (3, 5) + (8, 6) = (11, 11)

. Let's check each wrong answer. Choice A)

(7, 8)

gives

\vec{QR} = (4, 3)

, which equals

\vec{PQ}

rather than

2\vec{PQ}

– this misses the magnitude requirement. Choice C)

(5, 7)

gives

\vec{QR} = (2, 2)

, which has neither the correct direction nor magnitude. Choice D)

(-1, 2)

is actually point

P

, which would make

\vec{QR} = (-4, -3)

– this points in the opposite direction from

\vec{PQ}

. The correct answer is B)

(11, 11)

. Study tip: When working with collinear points and vector scaling, always verify both direction (same components or scalar multiples) and magnitude (use the distance formula). Draw a quick sketch to visualize the direction if needed.

Question 9

Point $A$ has position vector $\langle -2, 5 \rangle$ and point $B$ has position vector $\langle 4, 0 \rangle$ . In component form, the vector from $A$ to $B$ is represented as

$\langle 2, 5 \rangle$
$\langle 6, -5 \rangle$ (correct answer)
$\langle -6, 5 \rangle$
$\langle 4, 0 \rangle$

Explanation: This question tests understanding of how to find vector components by subtracting the coordinates of the initial point from the coordinates of the terminal point. A directed line segment from point A to point B represents a vector whose components are the differences in coordinates: ⟨final x - initial x, final y - initial y⟩. The horizontal component 4 - (-2) = 6 tells us the vector moves 6 units right, and the vertical component 0 - 5 = -5 tells us it moves -5 units down. Choice B is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨4 - (-2), 0 - 5⟩ = ⟨6, -5⟩. Choice C reverses the subtraction order, calculating ⟨x₁ - x₂, y₁ - y₂⟩, which gives the opposite vector (same magnitude, opposite direction). To avoid errors, clearly label which point is initial and which is terminal, then methodically compute x₂ - x₁ for the horizontal component and y₂ - y₁ for the vertical component. The negative sign in a component is meaningful: negative horizontal component means leftward motion, negative vertical component means downward motion.

Question 10

An object moves from position $(6, 1)$ to position $(2, 7)$ in the coordinate plane. Which vector correctly represents the displacement from the initial point to the terminal point?

$\langle 4, -6\rangle$
$\langle -4, 6\rangle$ (correct answer)
$\langle 8, 8\rangle$
$\langle 2, 7\rangle$

Explanation: This question tests understanding of how to find vector components by subtracting the coordinates of the initial point from the coordinates of the terminal point. The component form of a vector from point A(x₁, y₁) to point B(x₂, y₂) is found by subtracting the initial point's coordinates from the terminal point's coordinates: ⟨x₂ - x₁, y₂ - y₁⟩. For a vector from (6, 1) to (2, 7), we calculate the horizontal component as 2 - 6 = -4, and the vertical component as 7 - 1 = 6, giving us the component form ⟨-4, 6⟩. Choice B is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨2 - 6, 7 - 1⟩ = ⟨-4, 6⟩. Choice C adds the coordinates instead of subtracting them, which doesn't represent any meaningful vector operation for finding components. To avoid errors, clearly label which point is initial and which is terminal, then methodically compute x₂ - x₁ for the horizontal component and y₂ - y₁ for the vertical component. The negative sign in a component is meaningful: negative horizontal component means leftward motion, negative vertical component means downward motion.

Question 11

The vector $\overrightarrow{AB}$ goes from $A(-4, 0)$ to $B(-4, -3)$ . For the vector described, what is the component form of $\overrightarrow{AB}$ ?

$\langle 0, 3\rangle$
$\langle -3, 0\rangle$
$\langle 0, -3\rangle$ (correct answer)
$\langle -4, -3\rangle$

Explanation: This question tests understanding of how to find vector components by subtracting the coordinates of the initial point from the coordinates of the terminal point. A directed line segment from point A to point B represents a vector whose components are the differences in coordinates: ⟨final x - initial x, final y - initial y⟩. Using the formula for components, we substitute the given coordinates: ⟨x₂ - x₁, y₂ - y₁⟩ = ⟨-4 - (-4), -3 - 0⟩ = ⟨0, -3⟩. Choice C is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨-4 - (-4), -3 - 0⟩ = ⟨0, -3⟩. Choice A reverses the subtraction order, calculating ⟨x₁ - x₂, y₁ - y₂⟩, which gives the opposite vector (same magnitude, opposite direction). Key to finding vector components: always subtract initial point from terminal point (terminal - initial), and remember that the first component is the change in x, the second is the change in y. Check your work by visualizing: if point B is to the right of point A, the horizontal component should be positive; if B is above A, the vertical component should be positive.

Question 12

A vector has initial point $A(-2, 5)$ and terminal point $B(4, -1)$ . For the vector described, what is the component form of $\overrightarrow{AB}$ ?

$\langle -6, 6\rangle$
$\langle 4, -1\rangle$
$\langle 6, -6\rangle$ (correct answer)
$\langle 2, 4\rangle$

Explanation: This question tests understanding of how to find vector components by subtracting the coordinates of the initial point from the coordinates of the terminal point. The component form of a vector from point A(x₁, y₁) to point B(x₂, y₂) is found by subtracting the initial point's coordinates from the terminal point's coordinates: ⟨x₂ - x₁, y₂ - y₁⟩. For a vector from A(-2, 5) to B(4, -1), we calculate the horizontal component as 4 - (-2) = 6, and the vertical component as -1 - 5 = -6, giving us the component form ⟨6, -6⟩. Choice C is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨4 - (-2), -1 - 5⟩ = ⟨6, -6⟩. Choice A reverses the subtraction order, calculating ⟨x₁ - x₂, y₁ - y₂⟩, which gives the opposite vector (same magnitude, opposite direction). Key to finding vector components: always subtract initial point from terminal point (terminal - initial), and remember that the first component is the change in x, the second is the change in y. To avoid errors, clearly label which point is initial and which is terminal, then methodically compute x₂ - x₁ for the horizontal component and y₂ - y₁ for the vertical component.

Question 13

A vector has initial point $A(-4, 0)$ and terminal point $B(3, 2)$ . Based on the coordinates, what are the components of the vector from $A$ to $B$ ?

$\langle 7, 2 \rangle$ (correct answer)
$\langle -7, -2 \rangle$
$\langle -1, 2 \rangle$
$\langle 3, 2 \rangle$

Explanation: This question tests understanding of how to find vector components by subtracting the coordinates of the initial point from the coordinates of the terminal point. A directed line segment from point A to point B represents a vector whose components are the differences in coordinates: ⟨final x - initial x, final y - initial y⟩. Using the formula for components, we substitute the given coordinates: ⟨3 - (-4), 2 - 0⟩ = ⟨7, 2⟩. Choice A is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨3 - (-4), 2 - 0⟩ = ⟨7, 2⟩. Choice B reverses the subtraction order, calculating ⟨-4 - 3, 0 - 2⟩, which gives the opposite vector (same magnitude, opposite direction). Key to finding vector components: always subtract initial point from terminal point (terminal - initial), and remember that the first component is the change in x, the second is the change in y. To avoid errors, clearly label which point is initial and which is terminal, then methodically compute x₂ - x₁ for the horizontal component and y₂ - y₁ for the vertical component.

Question 14

Given points $A(2, 3)$ and $B(2, -4)$ , what is the component form of vector $\overrightarrow{AB}$ ?

$\langle 0, 7 \rangle$
$\langle 0, -7 \rangle$ (correct answer)
$\langle 4, -1 \rangle$
$\langle 2, -4 \rangle$

Explanation: This question tests understanding of how to find vector components by subtracting the coordinates of the initial point from the coordinates of the terminal point. The component form of a vector from point A(x₁, y₁) to point B(x₂, y₂) is found by subtracting the initial point's coordinates from the terminal point's coordinates: ⟨x₂ - x₁, y₂ - y₁⟩. For a vector from A(2, 3) to B(2, -4), we calculate the horizontal component as 2 - 2 = 0, and the vertical component as -4 - 3 = -7, giving us the component form ⟨0, -7⟩. Choice B is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨2 - 2, -4 - 3⟩ = ⟨0, -7⟩. Choice A reverses the subtraction order, calculating ⟨2 - 2, 3 - (-4)⟩, which gives the opposite vector (same magnitude, opposite direction). Check your work by visualizing: if there's no horizontal change, the horizontal component should be zero; if the terminal is below the initial, the vertical component should be negative. Remember that vector components are not the same as point coordinates—components represent displacement (change in position), while coordinates represent location.

Question 15

A vector has initial point $A(-6, 1)$ and terminal point $B(-2, 7)$ . What is the component form of $\overrightarrow{AB}$ ?

$\langle 4, 6 \rangle$ (correct answer)
$\langle -4, -6 \rangle$
$\langle -2, 7 \rangle$
$\langle -8, 8 \rangle$

Explanation: This question tests understanding of how to find vector components by subtracting the coordinates of the initial point from the coordinates of the terminal point. The component form of a vector from point A(x₁, y₁) to point B(x₂, y₂) is found by subtracting the initial point's coordinates from the terminal point's coordinates: ⟨x₂ - x₁, y₂ - y₁⟩. For a vector from A(-6, 1) to B(-2, 7), we calculate the horizontal component as -2 - (-6) = 4, and the vertical component as 7 - 1 = 6, giving us the component form ⟨4, 6⟩. Choice A is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨-2 - (-6), 7 - 1⟩ = ⟨4, 6⟩. Choice B reverses the subtraction order, calculating ⟨-6 - (-2), 1 - 7⟩, which gives the opposite vector (same magnitude, opposite direction). Key to finding vector components: always subtract initial point from terminal point (terminal - initial), and remember that the first component is the change in x, the second is the change in y. The negative sign in a component is meaningful: negative horizontal component means leftward motion, negative vertical component means downward motion.

Question 16

Given points $A(0, -2)$ (initial point) and $B(0, 5)$ (terminal point), what is the component form of $\overrightarrow{AB}$ ?

$\langle 0, 7\rangle$ (correct answer)
$\langle 7, 0\rangle$
$\langle 0, -7\rangle$
$\langle 0, 5\rangle$

Explanation: This question tests understanding of how to find vector components by subtracting the coordinates of the initial point from the coordinates of the terminal point. A directed line segment from point A to point B represents a vector whose components are the differences in coordinates: ⟨final x - initial x, final y - initial y⟩. Using the formula for components, we substitute the given coordinates: ⟨x₂ - x₁, y₂ - y₁⟩ = ⟨0 - 0, 5 - (-2)⟩ = ⟨0, 7⟩. Choice A is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨0 - 0, 5 - (-2)⟩ = ⟨0, 7⟩. Choice C reverses the subtraction order, calculating ⟨x₁ - x₂, y₁ - y₂⟩, which gives the opposite vector (same magnitude, opposite direction). Check your work by visualizing: if point B is to the right of point A, the horizontal component should be positive; if B is above A, the vertical component should be positive. In this case, both points have the same x-coordinate (0), so there's no horizontal displacement, but B is 7 units above A, giving a positive vertical component.

Question 17

Given points $A(5, 1)$ and $B(-1, 3)$ , what is the component form of vector $\overrightarrow{AB}$ ?

$\langle 4, 2 \rangle$
$\langle -6, 2 \rangle$ (correct answer)
$\langle -1, 3 \rangle$
$\langle 6, -2 \rangle$

Explanation: This question tests understanding of how to find vector components by subtracting the coordinates of the initial point from the coordinates of the terminal point. A directed line segment from point A to point B represents a vector whose components are the differences in coordinates: ⟨final x - initial x, final y - initial y⟩. The horizontal component -1 - 5 = -6 tells us the vector moves -6 units left, and the vertical component 3 - 1 = 2 tells us it moves 2 units up. Choice B is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨-1 - 5, 3 - 1⟩ = ⟨-6, 2⟩. Choice D adds the coordinates instead of subtracting them, which doesn't represent any meaningful vector operation for finding components. The negative sign in a component is meaningful: negative horizontal component means leftward motion, negative vertical component means downward motion. To avoid errors, clearly label which point is initial and which is terminal, then methodically compute x₂ - x₁ for the horizontal component and y₂ - y₁ for the vertical component.

Question 18

Given points $A(-4, -2)$ and $B(2, -2)$ , what are the components of the vector from $A$ to $B$ ?

$\langle 6, 0 \rangle$ (correct answer)
$\langle -6, 0 \rangle$
$\langle 0, 6 \rangle$
$\langle 2, -2 \rangle$

Explanation: This question tests understanding of how to find vector components by subtracting the coordinates of the initial point from the coordinates of the terminal point. The component form of a vector from point A(x₁, y₁) to point B(x₂, y₂) is found by subtracting the initial point's coordinates from the terminal point's coordinates: ⟨x₂ - x₁, y₂ - y₁⟩. For a vector from A(-4, -2) to B(2, -2), we calculate the horizontal component as 2 - (-4) = 6, and the vertical component as -2 - (-2) = 0, giving us the component form ⟨6, 0⟩. Choice A is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨2 - (-4), -2 - (-2)⟩ = ⟨6, 0⟩. Choice B reverses the subtraction order, calculating ⟨-4 - 2, -2 - (-2)⟩, which gives the opposite vector (same magnitude, opposite direction). Key to finding vector components: always subtract initial point from terminal point (terminal - initial), and remember that the first component is the change in x, the second is the change in y. Check your work by visualizing: if point B is to the right of point A, the horizontal component should be positive; if B is above A, the vertical component should be positive.

Question 19

Point $A$ is located at $(1, 6)$ and point $B$ is located at $(-3, 2)$ . For the vector described, what is the component form of $\overrightarrow{AB}$ ?

$\langle -4, -4\rangle$ (correct answer)
$\langle 4, 4\rangle$
$\langle -3, 2\rangle$
$\langle -4, 4\rangle$

Explanation: This question tests understanding of how to find vector components by subtracting the coordinates of the initial point from the coordinates of the terminal point. To find vector components, calculate the change in x (horizontal displacement) and the change in y (vertical displacement) by subtracting initial from terminal: horizontal component = x₂ - x₁, vertical component = y₂ - y₁. For a vector from A(1, 6) to B(-3, 2), we calculate the horizontal component as -3 - 1 = -4, and the vertical component as 2 - 6 = -4, giving us the component form ⟨-4, -4⟩. Choice A is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨-3 - 1, 2 - 6⟩ = ⟨-4, -4⟩. Choice B reverses the subtraction order, calculating ⟨x₁ - x₂, y₁ - y₂⟩, which gives the opposite vector (same magnitude, opposite direction). The negative sign in a component is meaningful: negative horizontal component means leftward motion, negative vertical component means downward motion. Remember that vector components are not the same as point coordinates—components represent displacement (change in position), while coordinates represent location.

Question 20

A vector has initial point $A(1, 6)$ and terminal point $B(1, -2)$ . For the vector described, what is the component form of $\overrightarrow{AB}$ ?

$\langle 0, 8 \rangle$
$\langle 0, -8 \rangle$ (correct answer)
$\langle 2, 4 \rangle$
$\langle 1, -2 \rangle$

Explanation: This question tests understanding of how to find vector components by subtracting the coordinates of the initial point from the coordinates of the terminal point. To find vector components, calculate the change in x (horizontal displacement) and the change in y (vertical displacement) by subtracting initial from terminal: horizontal component = x₂ - x₁, vertical component = y₂ - y₁. For a vector from A(1, 6) to B(1, -2), we calculate the horizontal component as 1 - 1 = 0, and the vertical component as -2 - 6 = -8, giving us the component form ⟨0, -8⟩. Choice B is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨1 - 1, -2 - 6⟩ = ⟨0, -8⟩. Choice A reverses the subtraction order, calculating ⟨x₁ - x₂, y₁ - y₂⟩, which gives the opposite vector (same magnitude, opposite direction). The negative sign in a component is meaningful: negative horizontal component means leftward motion, negative vertical component means downward motion. Check your work by visualizing: if point B is to the right of point A, the horizontal component should be positive; if B is above A, the vertical component should be positive.