Precalculus Quiz: Subtract And Represent Vectors Graphically
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Subtract And Represent Vectors GraphicallyQuestion 1 of 20

For vectors v⃗=⟨−2,6⟩\vec v=\langle -2, 6 \rangle and w⃗=⟨3,−1⟩\vec w=\langle 3, -1 \rangle, what is v⃗−w⃗\vec v-\vec w? Use v⃗−w⃗=v⃗+(−w⃗)\vec v-\vec w=\vec v+(-\vec w) and subtract corresponding components.

⟨1,5⟩\langle 1, 5 \rangle
⟨−5,7⟩\langle -5, 7 \rangle
⟨−6,−6⟩\langle -6, -6 \rangle
⟨5,−7⟩\langle 5, -7 \rangle
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Precalculus Quiz

Precalculus Quiz: Subtract And Represent Vectors Graphically

Practice Subtract And Represent Vectors Graphically 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 Subtract And Represent Vectors Graphically, giving you a quick way to practice the rules, question types, and explanations that matter most for Precalculus.

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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

For vectors v⃗=⟨−2,6⟩\vec v=\langle -2, 6 \rangle and w⃗=⟨3,−1⟩\vec w=\langle 3, -1 \rangle, what is v⃗−w⃗\vec v-\vec w? Use v⃗−w⃗=v⃗+(−w⃗)\vec v-\vec w=\vec v+(-\vec w) and subtract corresponding components.

  1. ⟨1,5⟩\langle 1, 5 \rangle
  2. ⟨−5,7⟩\langle -5, 7 \rangle (correct answer)
  3. ⟨−6,−6⟩\langle -6, -6 \rangle
  4. ⟨5,−7⟩\langle 5, -7 \rangle
Explanation: This question tests understanding of vector subtraction and how to represent it algebraically using components. Vector subtraction is defined as adding the opposite: v - w = v + (-w), where -w is the vector with the same magnitude as w but pointing in the opposite direction, having components -w = ⟨-w₁, -w₂⟩. For v = ⟨-2, 6⟩ and w = ⟨3, -1⟩, we compute v - w = ⟨-2 - 3, 6 - (-1)⟩ = ⟨-5, 7⟩. Choice B is correct because it applies the component subtraction formula correctly. Choice D reverses the subtraction order, computing w - v instead of v - w, which gives the opposite vector with all signs flipped. Key to vector subtraction: remember that v - w = v + (-w), so you can either subtract components directly (⟨v₁ - w₁, v₂ - w₂⟩) or add the opposite vector (-w), both methods give the same result. To check your work, verify that v = w + (v - w) using vector addition—if you add w back to your result v - w, you should get v.

Question 2

For vectors v⃗=⟨3,2⟩\vec v=\langle 3,2\rangle and w⃗=⟨−1,5⟩\vec w=\langle -1,5\rangle, how does v⃗−w⃗\vec v-\vec w compare to w⃗−v⃗\vec w-\vec v? (Use the idea that subtraction is adding the opposite.)

  1. v⃗−w⃗=w⃗−v⃗\vec v-\vec w=\vec w-\vec v for all vectors.
  2. v⃗−w⃗=−(w⃗−v⃗)\vec v-\vec w=-(\vec w-\vec v). (correct answer)
  3. v⃗−w⃗=v⃗+w⃗\vec v-\vec w=\vec v+\vec w.
  4. v⃗−w⃗=w⃗+v⃗\vec v-\vec w=\vec w+\vec v.
Explanation: This question tests understanding of vector subtraction and how to represent it as addition of the opposite. Vector subtraction is not commutative: v - w ≠ w - v in general; in fact, v - w = -(w - v), meaning they point in opposite directions with the same magnitude. Computing both: v - w = ⟨3 - (-1), 2 - 5⟩ = ⟨4, -3⟩ and w - v = ⟨-1 - 3, 5 - 2⟩ = ⟨-4, 3⟩, we see that w - v = -(v - w), confirming they have equal magnitudes but opposite directions. Choice B is correct because it correctly identifies the relationship between v - w and w - v as opposites. Choice A incorrectly assumes vector subtraction is commutative, claiming v - w = w - v, when in fact they are opposites. Remember that vector subtraction is not commutative: v - w and w - v are different vectors pointing in opposite directions, so order matters in subtraction unlike addition. To check your work, verify that v = w + (v - w) using vector addition—if you add w back to your result v - w, you should get v.

Question 3

For vectors v⃗=⟨−2,6⟩\vec v=\langle -2,6\rangle and w⃗=⟨3,−1⟩\vec w=\langle 3,-1\rangle, rewrite v⃗−w⃗\vec v-\vec w as v⃗+(−w⃗)\vec v+(-\vec w) and find the components of v⃗−w⃗\vec v-\vec w.

  1. ⟨1,5⟩\langle 1,5\rangle
  2. ⟨−5,7⟩\langle -5,7\rangle (correct answer)
  3. ⟨5,−7⟩\langle 5,-7\rangle
  4. ⟨−6,−6⟩\langle -6,-6\rangle
Explanation: This question tests understanding of vector subtraction and how to represent it algebraically using components. Vector subtraction is defined as adding the opposite: v - w = v + (-w), where -w is the vector with the same magnitude as w but pointing in the opposite direction, having components -w = ⟨-w₁, -w₂⟩. For v = ⟨-2, 6⟩ and w = ⟨3, -1⟩, we compute v - w = ⟨-2 - 3, 6 - (-1)⟩ = ⟨-5, 7⟩. Choice B is correct because it applies the component subtraction formula correctly. Choice C reverses the subtraction order, computing w - v instead of v - w, which gives the opposite vector with all signs flipped. Key to vector subtraction: remember that v - w = v + (-w), so you can either subtract components directly (⟨v₁ - w₁, v₂ - w₂⟩) or add the opposite vector (-w), both methods give the same result. To check your work, verify that v = w + (v - w) using vector addition—if you add w back to your result v - w, you should get v.

Question 4

Compare v⃗−w⃗\vec v-\vec w and w⃗−v⃗\vec w-\vec v for v⃗=⟨−1,4⟩\vec v=\langle -1, 4 \rangle and w⃗=⟨2,0⟩\vec w=\langle 2, 0 \rangle. Which statement is true?

  1. v⃗−w⃗=w⃗−v⃗\vec v-\vec w=\vec w-\vec v.
  2. v⃗−w⃗=−(w⃗−v⃗)\vec v-\vec w=-(\vec w-\vec v). (correct answer)
  3. v⃗−w⃗=v⃗+w⃗\vec v-\vec w=\vec v+\vec w.
  4. v⃗−w⃗\vec v-\vec w is always the zero vector.
Explanation: This question tests understanding of vector subtraction and how to represent it algebraically using components. Vector subtraction is defined as adding the opposite: v - w = v + (-w), where -w is the vector with the same magnitude as w but pointing in the opposite direction, having components -w = ⟨-w₁, -w₂⟩. Computing both: v - w = ⟨-1 - 2, 4 - 0⟩ = ⟨-3, 4⟩ and w - v = ⟨2 - (-1), 0 - 4⟩ = ⟨3, -4⟩, we see that w - v = -(v - w), confirming they have equal magnitudes but opposite directions. Choice B is correct because it correctly identifies the relationship. Choice A incorrectly assumes vector subtraction is commutative, claiming v - w = w - v, when in fact they are opposites. Remember that vector subtraction is not commutative: v - w and w - v are different vectors pointing in opposite directions, so order matters in subtraction unlike addition. To check your work, verify that v = w + (v - w) using vector addition—if you add w back to your result v - w, you should get v.

Question 5

Using the vectors given, what is −w⃗-\vec w if w⃗=⟨−7,3⟩\vec w=\langle -7,3\rangle? (This is the opposite vector used to rewrite v⃗−w⃗\vec v-\vec w as v⃗+(−w⃗)\vec v+(-\vec w).)

  1. ⟨7,3⟩\langle 7,3\rangle
  2. ⟨−7,−3⟩\langle -7,-3\rangle
  3. ⟨7,−3⟩\langle 7,-3\rangle (correct answer)
  4. ⟨−3,7⟩\langle -3,7\rangle
Explanation: This question tests understanding of vector subtraction and how to represent it as addition of the opposite. Vector subtraction is defined as adding the opposite: v - w = v + (-w), where -w is the vector with the same magnitude as w but pointing in the opposite direction, having components -w = ⟨-w₁, -w₂⟩. If w = ⟨-7, 3⟩, then -w = ⟨-(-7), -3⟩ = ⟨7, -3⟩, which means we negate both components; this gives a vector with the same length but pointing in the exact opposite direction. Choice C is correct because it correctly negates both components for -w. Choice B only negates one component when finding -w, but the opposite of a vector requires negating both components. When finding the opposite of a vector, -w = ⟨-w₁, -w₂⟩, negate every component; this creates a vector with identical magnitude but pointing in the exactly opposite direction (180° rotation). To check your work, verify that v = w + (v - w) using vector addition—if you add w back to your result v - w, you should get v.

Question 6

For vectors v⃗=⟨2,1⟩\vec v=\langle 2,1\rangle and w⃗=⟨5,3⟩\vec w=\langle 5,3\rangle, what is v⃗−w⃗\vec v-\vec w in component form? (Remember v⃗−w⃗=v⃗+(−w⃗)\vec v-\vec w=\vec v+(-\vec w).)

  1. ⟨7,4⟩\langle 7,4\rangle
  2. ⟨3,2⟩\langle 3,2\rangle
  3. ⟨−3,−2⟩\langle -3,-2\rangle (correct answer)
  4. ⟨−2,−3⟩\langle -2,-3\rangle
Explanation: This question tests understanding of vector subtraction and how to represent it algebraically using components. To subtract vectors component-wise, subtract the corresponding components: if v = ⟨v₁, v₂⟩ and w = ⟨w₁, w₂⟩, then v - w = ⟨v₁ - w₁, v₂ - w₂⟩. For v = ⟨2, 1⟩ and w = ⟨5, 3⟩, we compute v - w = ⟨2 - 5, 1 - 3⟩ = ⟨-3, -2⟩. Choice C is correct because it applies the component subtraction formula correctly, recognizing that when we subtract a larger vector from a smaller one, we get negative components. Choice B reverses the subtraction order, computing w - v instead of v - w, which gives ⟨5 - 2, 3 - 1⟩ = ⟨3, 2⟩. Remember that vector subtraction is not commutative: v - w and w - v are different vectors pointing in opposite directions, so order matters in subtraction unlike addition. To check your work, verify that v = w + (v - w) using vector addition—if you add w back to your result v - w, you should get v.

Question 7

Compare v⃗−w⃗\vec v-\vec w and w⃗−v⃗\vec w-\vec v for v⃗=⟨3,−2⟩\vec v=\langle 3,-2\rangle and w⃗=⟨−1,4⟩\vec w=\langle -1,4\rangle. Which statement is true? (Use the idea that subtraction is adding the opposite.)

  1. v⃗−w⃗=w⃗−v⃗\vec v-\vec w=\vec w-\vec v for all vectors.
  2. v⃗−w⃗=−(w⃗−v⃗)\vec v-\vec w=-(\vec w-\vec v). (correct answer)
  3. v⃗−w⃗=v⃗+w⃗\vec v-\vec w=\vec v+\vec w.
  4. v⃗−w⃗\vec v-\vec w is always the zero vector.
Explanation: This question tests understanding of vector subtraction and how to represent it as addition of the opposite. Vector subtraction is not commutative: v - w ≠ w - v in general; in fact, v - w = -(w - v), meaning they point in opposite directions with the same magnitude. Computing both: v - w = ⟨3 - (-1), -2 - 4⟩ = ⟨4, -6⟩ and w - v = ⟨-1 - 3, 4 - (-2)⟩ = ⟨-4, 6⟩, we see that w - v = -(v - w), confirming they have equal magnitudes but opposite directions. Choice B is correct because it correctly identifies the relationship. Choice A incorrectly assumes vector subtraction is commutative, claiming v - w = w - v, when in fact they are opposites. Remember that vector subtraction is not commutative: v - w and w - v are different vectors pointing in opposite directions, so order matters in subtraction unlike addition. To check your work, verify that v = w + (v - w) using vector addition—if you add w back to your result v - w, you should get v.

Question 8

Compare v⃗−w⃗\vec v-\vec w and w⃗−v⃗\vec w-\vec v for v⃗=⟨2,1⟩\vec v=\langle 2,1\rangle and w⃗=⟨5,3⟩\vec w=\langle 5,3\rangle. Which statement is true? (Think of subtraction as adding the opposite.)

  1. v⃗−w⃗=w⃗−v⃗\vec v-\vec w=\vec w-\vec v for all vectors.
  2. v⃗−w⃗=−(w⃗−v⃗)\vec v-\vec w=-(\vec w-\vec v). (correct answer)
  3. v⃗−w⃗=v⃗+w⃗\vec v-\vec w=\vec v+\vec w.
  4. v⃗−w⃗=⟨∣2−5∣,∣1−3∣⟩\vec v-\vec w=\langle |2-5|,|1-3|\rangle.
Explanation: This question tests understanding of vector subtraction and how to represent it as addition of the opposite. Vector subtraction is defined as adding the opposite: v - w = v + (-w), where -w is the vector with the same magnitude as w but pointing in the opposite direction, having components -w = ⟨-w₁, -w₂⟩. Computing both: v - w = ⟨2 - 5, 1 - 3⟩ = ⟨-3, -2⟩ and w - v = ⟨5 - 2, 3 - 1⟩ = ⟨3, 2⟩, we see that w - v = -(v - w), confirming they have equal magnitudes but opposite directions. Choice B is correct because it correctly identifies the relationship. Choice A incorrectly assumes vector subtraction is commutative, claiming v - w = w - v, when in fact they are opposites. Remember that vector subtraction is not commutative: v - w and w - v are different vectors pointing in opposite directions, so order matters in subtraction unlike addition. To check your work, verify that v = w + (v - w) using vector addition—if you add w back to your result v - w, you should get v.

Question 9

For vectors v⃗=⟨0,−3⟩\vec v=\langle 0,-3\rangle and w⃗=⟨−2,4⟩\vec w=\langle -2,4\rangle, what is v⃗−w⃗\vec v-\vec w? (Equivalently, compute v⃗+(−w⃗)\vec v+(-\vec w).)

  1. ⟨−2,1⟩\langle -2,1\rangle
  2. ⟨2,−7⟩\langle 2,-7\rangle (correct answer)
  3. ⟨−2,−7⟩\langle -2,-7\rangle
  4. ⟨2,7⟩\langle 2,7\rangle
Explanation: This question tests understanding of vector subtraction and how to represent it as addition of the opposite. Vector subtraction is defined as adding the opposite: v - w = v + (-w), where -w is the vector with the same magnitude as w but pointing in the opposite direction, having components -w = ⟨-w₁, -w₂⟩. For v = ⟨0, -3⟩ and w = ⟨-2, 4⟩, we compute v - w = ⟨0 - (-2), -3 - 4⟩ = ⟨2, -7⟩. Choice B is correct because it applies the component subtraction formula correctly, properly handling the double negative in the first component. Choice C makes an arithmetic error in the subtraction, calculating ⟨-2, -7⟩ instead of ⟨2, -7⟩, likely by incorrectly handling the subtraction of the negative first component. Key to vector subtraction: remember that v - w = v + (-w), so you can either subtract components directly (⟨v₁ - w₁, v₂ - w₂⟩) or add the opposite vector (-w), both methods give the same result. To check your work, verify that v = w + (v - w) using vector addition—if you add w back to your result v - w, you should get v.

Question 10

Vector u⃗=⟨5,−2⟩\vec{u} = \langle 5, -2 \rangle and vector v⃗=⟨−3,4⟩\vec{v} = \langle -3, 4 \rangle. If the resultant vector w⃗=u⃗−v⃗\vec{w} = \vec{u} - \vec{v} is represented graphically starting from the origin, what are the coordinates of its terminal point?

  1. ⟨8,−6⟩\langle 8, -6 \rangle (correct answer)
  2. ⟨2,2⟩\langle 2, 2 \rangle
  3. ⟨−8,6⟩\langle -8, 6 \rangle
  4. ⟨−2,−2⟩\langle -2, -2 \rangle
Explanation: Vector subtraction is performed component-wise: u⃗−v⃗=⟨5,−2⟩−⟨−3,4⟩=⟨5−(−3),−2−4⟩=⟨8,−6⟩\vec{u} - \vec{v} = \langle 5, -2 \rangle - \langle -3, 4 \rangle = \langle 5-(-3), -2-4 \rangle = \langle 8, -6 \rangle. Choice B results from adding the vectors instead of subtracting. Choice C results from computing v⃗−u⃗\vec{v} - \vec{u} instead of u⃗−v⃗\vec{u} - \vec{v}. Choice D results from both reversing the order and making sign errors.

Question 11

If p⃗=⟨7,−1⟩\vec{p} = \langle 7, -1 \rangle and q⃗=⟨2,5⟩\vec{q} = \langle 2, 5 \rangle, what is the magnitude of p⃗−q⃗\vec{p} - \vec{q}?

  1. 61\sqrt{61} (correct answer)
  2. 45\sqrt{45}
  3. 85\sqrt{85}
  4. 29\sqrt{29}
Explanation: First compute p⃗−q⃗=⟨7,−1⟩−⟨2,5⟩=⟨5,−6⟩\vec{p} - \vec{q} = \langle 7, -1 \rangle - \langle 2, 5 \rangle = \langle 5, -6 \rangle. Then find the magnitude: ∣p⃗−q⃗∣=52+(−6)2=25+36=61|\vec{p} - \vec{q}| = \sqrt{5^2 + (-6)^2} = \sqrt{25 + 36} = \sqrt{61}. Choice B results from computing 32+62\sqrt{3^2 + 6^2} (wrong x-component). Choice C results from computing ∣p⃗+q⃗∣|\vec{p} + \vec{q}| instead. Choice D results from computing ∣q⃗−p⃗∣|\vec{q} - \vec{p}| with a calculation error.

Question 12

For vectors v⃗=⟨5,3⟩\vec v=\langle 5,3\rangle and w⃗=⟨2,1⟩\vec w=\langle 2,1\rangle, what are the components of v⃗−w⃗\vec v-\vec w? (Recall: vector subtraction means adding the opposite, v⃗−w⃗=v⃗+(−w⃗)\vec v-\vec w=\vec v+(-\vec w).)

  1. ⟨7,4⟩\langle 7,4\rangle
  2. ⟨3,2⟩\langle 3,2\rangle (correct answer)
  3. ⟨−3,−2⟩\langle -3,-2\rangle
  4. ⟨10,3⟩\langle 10,3\rangle
Explanation: This question tests understanding of vector subtraction and how to represent it algebraically using components. Vector subtraction is defined as adding the opposite: v - w = v + (-w), where -w is the vector with the same magnitude as w but pointing in the opposite direction, having components -w = ⟨-w₁, -w₂⟩. For v = ⟨5, 3⟩ and w = ⟨2, 1⟩, we compute v - w = ⟨5 - 2, 3 - 1⟩ = ⟨3, 2⟩. Choice B is correct because it applies the component subtraction formula correctly, giving ⟨3, 2⟩. Choice A adds the vectors instead of subtracting them, calculating v + w = ⟨5 + 2, 3 + 1⟩ = ⟨7, 4⟩ when we need v - w. Key to vector subtraction: remember that v - w = v + (-w), so you can either subtract components directly (⟨v₁ - w₁, v₂ - w₂⟩) or add the opposite vector (-w), both methods give the same result. To check your work, verify that v = w + (v - w) using vector addition—if you add w back to your result v - w, you should get v.

Question 13

If w⃗=⟨−4,3⟩\vec w=\langle -4, 3 \rangle, what is the opposite vector −w⃗-\vec w (the vector with the same magnitude but opposite direction), which is used when rewriting v⃗−w⃗\vec v-\vec w as v⃗+(−w⃗)\vec v+(-\vec w)?

  1. ⟨4,−3⟩\langle 4, -3 \rangle (correct answer)
  2. ⟨−4,−3⟩\langle -4, -3 \rangle
  3. ⟨4,3⟩\langle 4, 3 \rangle
  4. ⟨−3,4⟩\langle -3, 4 \rangle
Explanation: This question tests understanding of vector subtraction and how to represent it as addition of the opposite. Vector subtraction is defined as adding the opposite: v - w = v + (-w), where -w is the vector with the same magnitude as w but pointing in the opposite direction, having components -w = ⟨-w₁, -w₂⟩. If w = ⟨-4, 3⟩, then -w = ⟨-(-4), -3⟩ = ⟨4, -3⟩, which means we negate both components; this gives a vector with the same length but pointing in the exact opposite direction. Choice A is correct because it correctly negates both components for -w. Choice B only negates one component when finding -w, but the opposite of a vector requires negating both components. When finding the opposite of a vector, -w = ⟨-w₁, -w₂⟩, negate every component; this creates a vector with identical magnitude but pointing in the exactly opposite direction (180° rotation). To check your work, verify that v = w + (v - w) using vector addition—if you add w back to your result v - w, you should get v.

Question 14

An object's displacement over two steps is represented by vectors v⃗=⟨6,0⟩\vec v=\langle 6,0\rangle and w⃗=⟨1,0⟩\vec w=\langle 1,0\rangle drawn from the same starting point. The difference v⃗−w⃗\vec v-\vec w can be found as v⃗+(−w⃗)\vec v+(-\vec w). What are the components of v⃗−w⃗\vec v-\vec w?

  1. ⟨7,0⟩\langle 7,0\rangle
  2. ⟨5,0⟩\langle 5,0\rangle (correct answer)
  3. ⟨−5,0⟩\langle -5,0\rangle
  4. ⟨6,1⟩\langle 6,1\rangle
Explanation: This question tests understanding of vector subtraction and how to represent it algebraically using components. Vector subtraction is defined as adding the opposite: v - w = v + (-w), where -w is the vector with the same magnitude as w but pointing in the opposite direction, having components -w = ⟨-w₁, -w₂⟩. For v = ⟨6, 0⟩ and w = ⟨1, 0⟩, we compute v - w = ⟨6 - 1, 0 - 0⟩ = ⟨5, 0⟩. Choice B is correct because it applies the component subtraction formula correctly. Choice C reverses the subtraction order, computing w - v instead of v - w, which gives the opposite vector with all signs flipped. Key to vector subtraction: remember that v - w = v + (-w), so you can either subtract components directly (⟨v₁ - w₁, v₂ - w₂⟩) or add the opposite vector (-w), both methods give the same result. To check your work, verify that v = w + (v - w) using vector addition—if you add w back to your result v - w, you should get v.

Question 15

For vectors v⃗=⟨5,3⟩\vec v=\langle 5,3\rangle and w⃗=⟨2,1⟩\vec w=\langle 2,1\rangle, use component-wise subtraction (equivalently, v⃗−w⃗=v⃗+(−w⃗)\vec v-\vec w=\vec v+(-\vec w)) to find the components of v⃗−w⃗\vec v-\vec w in ⟨a,b⟩\langle a,b\rangle form.

  1. ⟨7,4⟩\langle 7,4\rangle
  2. ⟨3,2⟩\langle 3,2\rangle (correct answer)
  3. ⟨−3,−2⟩\langle -3,-2\rangle
  4. ⟨10,3⟩\langle 10,3\rangle
Explanation: This question tests understanding of vector subtraction and how to represent it algebraically using components. To subtract vectors component-wise, subtract the corresponding components: if v = ⟨v₁, v₂⟩ and w = ⟨w₁, w₂⟩, then v - w = ⟨v₁ - w₁, v₂ - w₂⟩. For v = ⟨5, 3⟩ and w = ⟨2, 1⟩, we compute v - w = ⟨5 - 2, 3 - 1⟩ = ⟨3, 2⟩. Choice B is correct because it applies the component subtraction formula correctly. Choice A adds the vectors instead of subtracting them, calculating v + w = ⟨5 + 2, 3 + 1⟩ = ⟨7, 4⟩ when we need v - w. Key to vector subtraction: remember that v - w = v + (-w), so you can either subtract components directly (⟨v₁ - w₁, v₂ - w₂⟩) or add the opposite vector (-w), both methods give the same result. To check your work, verify that v = w + (v - w) using vector addition—if you add w back to your result v - w, you should get v.

Question 16

Vector m⃗\vec{m} has initial point (3,−2)(3, -2) and terminal point (8,−5)(8, -5). Vector n⃗\vec{n} has initial point (−1,4)(-1, 4) and terminal point (2,6)(2, 6). What is m⃗−n⃗\vec{m} - \vec{n} expressed in component form?

  1. ⟨6,1⟩\langle 6, 1 \rangle
  2. ⟨8,−3⟩\langle 8, -3 \rangle
  3. ⟨−2,5⟩\langle -2, 5 \rangle
  4. ⟨2,−5⟩\langle 2, -5 \rangle (correct answer)
Explanation: When working with vectors defined by initial and terminal points, you need to find the component form of each vector before performing any operations. The component form of a vector is found by subtracting the coordinates of the initial point from the coordinates of the terminal point. For vector m⃗\vec{m} with initial point (3,−2)(3, -2) and terminal point (8,−5)(8, -5): m⃗=⟨8−3,−5−(−2)⟩=⟨5,−3⟩\vec{m} = \langle 8-3, -5-(-2) \rangle = \langle 5, -3 \rangle For vector n⃗\vec{n} with initial point (−1,4)(-1, 4) and terminal point (2,6)(2, 6): n⃗=⟨2−(−1),6−4⟩=⟨3,2⟩\vec{n} = \langle 2-(-1), 6-4 \rangle = \langle 3, 2 \rangle Now you can find m⃗−n⃗\vec{m} - \vec{n} by subtracting corresponding components: m⃗−n⃗=⟨5,−3⟩−⟨3,2⟩=⟨5−3,−3−2⟩=⟨2,−5⟩\vec{m} - \vec{n} = \langle 5, -3 \rangle - \langle 3, 2 \rangle = \langle 5-3, -3-2 \rangle = \langle 2, -5 \rangle This confirms answer choice D is correct. Looking at the wrong answers: Choice A, ⟨6,1⟩\langle 6, 1 \rangle, likely comes from adding the vectors instead of subtracting. Choice B, ⟨8,−3⟩\langle 8, -3 \rangle, appears to use just the terminal point of m⃗\vec{m} without proper vector subtraction. Choice C, ⟨−2,5⟩\langle -2, 5 \rangle, results from calculating n⃗−m⃗\vec{n} - \vec{m} instead of m⃗−n⃗\vec{m} - \vec{n}. Remember: always convert position vectors to component form first (terminal minus initial), then perform the vector operations component-wise. Vector subtraction is not commutative, so order matters.

Question 17

Using the vectors given, v⃗=⟨0,−4⟩\vec v=\langle 0, -4 \rangle and w⃗=⟨−3,2⟩\vec w=\langle -3, 2 \rangle, what are the components of v⃗−w⃗\vec v-\vec w? (Remember: subtracting w⃗\vec w is the same as adding the opposite vector −w⃗-\vec w.)

  1. ⟨−3,−2⟩\langle -3, -2 \rangle
  2. ⟨3,−6⟩\langle 3, -6 \rangle (correct answer)
  3. ⟨−3,−6⟩\langle -3, -6 \rangle
  4. ⟨3,6⟩\langle 3, 6 \rangle
Explanation: This question tests understanding of vector subtraction and how to represent it algebraically using components. Vector subtraction is defined as adding the opposite: v - w = v + (-w), where -w is the vector with the same magnitude as w but pointing in the opposite direction, having components -w = ⟨-w₁, -w₂⟩. For v = ⟨0, -4⟩ and w = ⟨-3, 2⟩, we compute v - w = ⟨0 - (-3), -4 - 2⟩ = ⟨3, -6⟩. Choice B is correct because it applies the component subtraction formula correctly. Choice A reverses the subtraction order, computing w - v instead of v - w, which gives the opposite vector with all signs flipped. Key to vector subtraction: remember that v - w = v + (-w), so you can either subtract components directly (⟨v₁ - w₁, v₂ - w₂⟩) or add the opposite vector (-w), both methods give the same result. To check your work, verify that v = w + (v - w) using vector addition—if you add w back to your result v - w, you should get v.

Question 18

For vectors v⃗\vec v and w⃗\vec w drawn tail-to-tail from the same point, which statement correctly describes the graphical representation of v⃗−w⃗\vec v-\vec w using the tip-to-tip method (and the idea v⃗−w⃗=v⃗+(−w⃗)\vec v-\vec w=\vec v+(-\vec w))?

  1. v⃗−w⃗\vec v-\vec w is the vector from the tip of v⃗\vec v to the tip of w⃗\vec w.
  2. v⃗−w⃗\vec v-\vec w is the vector from the tail of v⃗\vec v to the tail of w⃗\vec w.
  3. v⃗−w⃗\vec v-\vec w is the vector from the tip of w⃗\vec w to the tip of v⃗\vec v. (correct answer)
  4. v⃗−w⃗\vec v-\vec w is the same as w⃗−v⃗\vec w-\vec v because subtraction is commutative.
Explanation: This question tests understanding of vector subtraction and how to represent it graphically using the tip-to-tip method. Graphically, when vectors v and w are drawn from the same initial point (tail-to-tail), the vector v - w is represented by the vector from the tip of w to the tip of v, forming a triangle. When v and w are drawn with their tails at the same point, visualize walking from the tip of w to the tip of v—that path represents v - w. This is equivalent to the triangle rule: v, w, and v - w form a closed triangle. Choice C is correct because it properly describes the tip-to-tip method. Choice A describes the graphical vector going from the tip of v to the tip of w, but v - w goes from the tip of w to the tip of v (the opposite direction). For the tip-to-tip graphical method, place both vectors tail-to-tail and draw the difference vector from the tip of what you're subtracting (w) to the tip of what you're subtracting from (v). Remember that vector subtraction is not commutative: v - w and w - v are different vectors pointing in opposite directions, so order matters in subtraction unlike addition.

Question 19

For vectors drawn tail-to-tail at the origin, v⃗=⟨4,0⟩\vec v = \langle 4, 0\rangle and w⃗=⟨1,0⟩\vec w = \langle 1, 0\rangle. Using v⃗−w⃗=v⃗+(−w⃗)\vec v-\vec w=\vec v+(-\vec w), what are the components of v⃗−w⃗\vec v-\vec w?

  1. ⟨3,0⟩\langle 3, 0\rangle (correct answer)
  2. ⟨5,0⟩\langle 5, 0\rangle
  3. ⟨−3,0⟩\langle -3, 0\rangle
  4. ⟨4,1⟩\langle 4, 1\rangle
Explanation: This question tests understanding of vector subtraction and how to represent it as addition of the opposite. Vector subtraction is defined as adding the opposite: v - w = v + (-w), where -w is the vector with the same magnitude as w but pointing in the opposite direction, having components -w = ⟨-w₁, -w₂⟩. For v = ⟨4, 0⟩ and w = ⟨1, 0⟩, we compute v - w = ⟨4 - 1, 0 - 0⟩ = ⟨3, 0⟩. Choice A is correct because it applies the component subtraction formula correctly, giving ⟨3, 0⟩. Choice B adds the vectors instead of subtracting them, calculating v + w = ⟨5, 0⟩ when we need v - w. To check your work, verify that v = w + (v - w) using vector addition—if you add w back to your result v - w, you should get v: ⟨1, 0⟩ + ⟨3, 0⟩ = ⟨4, 0⟩ ✓.

Question 20

Two vectors a⃗=⟨4,−3⟩\vec{a} = \langle 4, -3 \rangle and b⃗=⟨−1,5⟩\vec{b} = \langle -1, 5 \rangle are given. A student claims that a⃗−b⃗\vec{a} - \vec{b} has the same magnitude as b⃗−a⃗\vec{b} - \vec{a}. Which statement best evaluates this claim?

  1. The claim is correct because subtraction of vectors is commutative under magnitude calculations
  2. The claim is correct because a⃗−b⃗\vec{a} - \vec{b} and b⃗−a⃗\vec{b} - \vec{a} are additive inverses with equal magnitudes (correct answer)
  3. The claim is incorrect because vector subtraction reverses both direction and magnitude relationships
  4. The claim is incorrect because the components have different signs in each subtraction operation
Explanation: Computing both: a⃗−b⃗=⟨5,−8⟩\vec{a} - \vec{b} = \langle 5, -8 \rangle and b⃗−a⃗=⟨−5,8⟩\vec{b} - \vec{a} = \langle -5, 8 \rangle. These are additive inverses (opposite vectors) and therefore have the same magnitude: 25+64=89\sqrt{25 + 64} = \sqrt{89}. The claim is correct. Choice A incorrectly describes subtraction as commutative. Choice C incorrectly states that magnitude is affected by direction. Choice D focuses on signs rather than the magnitude relationship.