Finding Components of Vectors - Precalculus

Given points $A(-2, 1)$ and $B(0, -6)$ , what are the components of the vector from $A$ to $B$ ?
Answer: $\langle 2, -7 \rangle$ 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, 1) to B(0, -6), we calculate the horizontal component as 0 - (-2) = 2, and the vertical component as -6 - 1 = -7, giving us the component form ⟨2, -7⟩. Choice A is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨0 - (-2), -6 - 1⟩ = ⟨2, -7⟩. Choice B reverses the subtraction order, calculating ⟨-2 - 0, 1 - (-6)⟩, 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.
The directed line segment from $P(5, 0)$ to $Q(-1, 3)$ represents vector $\vec{v}$ . Based on the coordinates, what are the components of $\vec{v}$ ?
Answer: $\langle -6, 3\rangle$ 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 P(5, 0) to Q(-1, 3), we calculate the horizontal component as -1 - 5 = -6, and the vertical component as 3 - 0 = 3, giving us the component form ⟨-6, 3⟩. Choice B is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨-1 - 5, 3 - 0⟩ = ⟨-6, 3⟩. Choice C 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. The negative sign in a component is meaningful: negative horizontal component means leftward motion, negative vertical component means downward motion.
The directed line segment from $P(0, -3)$ to $Q(5, -3)$ represents a vector $\vec{v}$ . In component form, $\vec{v}$ is represented as:
Answer: $\langle 5, 0 \rangle$ 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 ⟨0 - 5, -3 - (-3)⟩, 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. 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.
A vector has initial point $A(5, -1)$ and terminal point $B(0, -6)$ . Based on the coordinates, what is the component form of $\overrightarrow{AB}$ ?
Answer: $\langle -5, -5\rangle$ 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(5, -1) to B(0, -6), we calculate the horizontal component as 0 - 5 = -5, and the vertical component as -6 - (-1) = -5, giving us the component form ⟨-5, -5⟩. Choice B is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨0 - 5, -6 - (-1)⟩ = ⟨-5, -5⟩. Choice A 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.
A vector has initial point $A(-3, 4)$ and terminal point $B(2, -1)$ . For the vector described, what is the component form of $\overrightarrow{AB}$ ?
Answer: $\langle 5, -5 \rangle$ 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( $x_1$ , $y_1$ ) to point B( $x_2$ , $y_2$ ) is found by subtracting the initial point's coordinates from the terminal point's coordinates: $\langle x_2 - x_1, y_2 - y_1 \rangle$ . For a vector from A(-3, 4) to B(2, -1), we calculate the horizontal component as $2 - (-3) = 5$ , and the vertical component as $-1 - 4 = -5$ , giving us the component form $\langle 5, -5 \rangle$ . Choice B is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: $\langle 2 - (-3), -1 - 4 \rangle = \langle 5, -5 \rangle$ . Choice A reverses the subtraction order, calculating $\langle -3 - 2, 4 - (-1) \rangle$ , 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 $x_2$ - $x_1$ for the horizontal component and $y_2$ - $y_1$ for the vertical component.
A vector has initial point $A(5, -1)$ and terminal point $B(0, 3)$ . For the vector described, what is the component form of $\overrightarrow{AB}$ ?
Answer: $\langle -5, 4 \rangle$ 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.
A vector has initial point $A(-6, 1)$ and terminal point $B(-2, 7)$ . What is the component form of $\overrightarrow{AB}$ ?
Answer: $\langle 4, 6 \rangle$ 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.
Given points $A(3, -4)$ and $B(-1, 2)$ , what are the components of the vector from $A$ to $B$ ?
Answer: $\langle -4, 6 \rangle$ 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: ⟨-1 - 3, 2 - (-4)⟩ = ⟨-4, 6⟩. Choice C is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨-1 - 3, 2 - (-4)⟩ = ⟨-4, 6⟩. Choice D reverses the subtraction order, calculating ⟨3 - (-1), -4 - 2⟩, 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.
An object moves from position $(-2, -3)$ to position $(-2, 4)$ in the coordinate plane. Which vector correctly represents the displacement from the initial point to the terminal point?
Answer: $\langle 0, 7 \rangle$ 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 (-2, -3) to (-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 A is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨-2 - (-2), 4 - (-3)⟩ = ⟨0, 7⟩. Choice B reverses the subtraction order, calculating ⟨-2 - (-2), -3 - 4⟩, 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 there's no horizontal change, the horizontal component should be zero; if the terminal is above the initial, the vertical component should be positive.
Point $A$ is located at $( -1, 4)$ and point $B$ is located at $(6, 1)$ . What is the component form of vector $\overrightarrow{AB}$ ?
Answer: $\langle 7, -3 \rangle$ 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 6 - (-1) = 7 tells us the vector moves 7 units right, and the vertical component 1 - 4 = -3 tells us it moves 3 units down. Choice A is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨6 - (-1), 1 - 4⟩ = ⟨7, -3⟩. Choice B reverses the subtraction order, calculating ⟨-1 - 6, 4 - 1⟩, 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. 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.
A vector has initial point $A(-4, 1)$ and terminal point $B(3, 1)$ . Based on the coordinates, what are the components of the vector from $A$ to $B$ ?
Answer: $\langle 7, 0\rangle$ 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⟩ = ⟨3 - (-4), 1 - 1⟩ = ⟨7, 0⟩. Choice B is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨3 - (-4), 1 - 1⟩ = ⟨7, 0⟩. Choice A reverses the subtraction order, calculating ⟨x₁ - x₂, y₁ - y₂⟩, which gives the opposite vector (same magnitude, opposite direction). The horizontal component 7 tells us the vector moves 7 units right, and the vertical component 0 tells us it moves 0 units up/down. 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.
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?
Answer: $(2.5, -1)$ 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.
Given points $A(5, 1)$ and $B(-1, 6)$ in the coordinate plane, what is the component form of vector $\overrightarrow{AB}$ ?
Answer: $\langle -6, 5 \rangle$ 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: ⟨-1 - 5, 6 - 1⟩ = ⟨-6, 5⟩. Choice B is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨-1 - 5, 6 - 1⟩ = ⟨-6, 5⟩. Choice C has the correct horizontal component but reverses the sign on the vertical component, mixing up the subtraction direction. 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.
Given points $A(-4, -2)$ and $B(2, -2)$ , what are the components of the vector from $A$ to $B$ ?
Answer: $\langle 6, 0 \rangle$ 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.
The directed line segment from $P(3, -4)$ to $Q(-1, 2)$ represents vector $\vec{v}$ . Based on the coordinates, what are the components of $\vec{v}$ ?
Answer: $\langle -4, 6\rangle$ 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: ⟨x₂ - x₁, y₂ - y₁⟩ = ⟨-1 - 3, 2 - (-4)⟩ = ⟨-4, 6⟩. Choice B is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨-1 - 3, 2 - (-4)⟩ = ⟨-4, 6⟩. Choice A reverses the subtraction order, calculating ⟨x₁ - x₂, y₁ - y₂⟩, 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.
A vector has initial point $A(-2, 5)$ and terminal point $B(4, -1)$ . For the vector described, what is the component form of $\overrightarrow{AB}$ ?
Answer: $\langle 6, -6\rangle$ 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.
The directed line segment from $P(0, -2)$ to $Q(6, -2)$ represents a vector $\vec{v}$ . What are the components of $\vec{v}$ ?
Answer: $\langle 6, 0 \rangle$ 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 P(0, -2) to Q(6, -2), we calculate the horizontal component as 6 - 0 = 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: ⟨6 - 0, -2 - (-2)⟩ = ⟨6, 0⟩. Choice B reverses the subtraction order, calculating ⟨x₁ - x₂, y₁ - y₂⟩, 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. 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.
Given points $A(-1, -5)$ (initial point) and $B(2, 4)$ (terminal point), what is the component form of $\overrightarrow{AB}$ ?
Answer: $\langle 3, 9\rangle$ 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(-1, -5) to B(2, 4), we calculate the horizontal component as 2 - (-1) = 3, and the vertical component as 4 - (-5) = 9, giving us the component form ⟨3, 9⟩. Choice A is correct because it properly subtracts the initial point coordinates from the terminal point coordinates: ⟨2 - (-1), 4 - (-5)⟩ = ⟨3, 9⟩. Choice B 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. 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 vector $\overrightarrow{AB}$ goes from $A(-4, 0)$ to $B(-4, -3)$ . For the vector described, what is the component form of $\overrightarrow{AB}$ ?
Answer: $\langle 0, -3\rangle$ 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.
A vector has initial point $A(1, 6)$ and terminal point $B(1, -2)$ . For the vector described, what is the component form of $\overrightarrow{AB}$ ?
Answer: $\langle 0, -8 \rangle$ 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.

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