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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.

Question 1 / 20

0 of 20 answered

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

Select an answer to continue

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.

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

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

  1. ⟨8,−6⟩\langle 8, -6 \rangle⟨8,−6⟩ (correct answer)
  2. ⟨2,2⟩\langle 2, 2 \rangle⟨2,2⟩
  3. ⟨−8,6⟩\langle -8, 6 \rangle⟨−8,6⟩
  4. ⟨−2,−2⟩\langle -2, -2 \rangle⟨−2,−2⟩

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 \rangleu−v=⟨5,−2⟩−⟨−3,4⟩=⟨5−(−3),−2−4⟩=⟨8,−6⟩. Choice B results from adding the vectors instead of subtracting. Choice C results from computing v⃗−u⃗\vec{v} - \vec{u}v−u instead of u⃗−v⃗\vec{u} - \vec{v}u−v. Choice D results from both reversing the order and making sign errors.

Question 2

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

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

Question 3

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

  1. ⟨7,4⟩\langle 7,4\rangle⟨7,4⟩
  2. ⟨3,2⟩\langle 3,2\rangle⟨3,2⟩ (correct answer)
  3. ⟨−3,−2⟩\langle -3,-2\rangle⟨−3,−2⟩
  4. ⟨10,3⟩\langle 10,3\rangle⟨10,3⟩

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 4

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

  1. ⟨4,−3⟩\langle 4, -3 \rangle⟨4,−3⟩ (correct answer)
  2. ⟨−4,−3⟩\langle -4, -3 \rangle⟨−4,−3⟩
  3. ⟨4,3⟩\langle 4, 3 \rangle⟨4,3⟩
  4. ⟨−3,4⟩\langle -3, 4 \rangle⟨−3,4⟩

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 5

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

  1. ⟨7,0⟩\langle 7,0\rangle⟨7,0⟩
  2. ⟨5,0⟩\langle 5,0\rangle⟨5,0⟩ (correct answer)
  3. ⟨−5,0⟩\langle -5,0\rangle⟨−5,0⟩
  4. ⟨6,1⟩\langle 6,1\rangle⟨6,1⟩

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 6

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

  1. ⟨7,4⟩\langle 7,4\rangle⟨7,4⟩
  2. ⟨3,2⟩\langle 3,2\rangle⟨3,2⟩ (correct answer)
  3. ⟨−3,−2⟩\langle -3,-2\rangle⟨−3,−2⟩
  4. ⟨10,3⟩\langle 10,3\rangle⟨10,3⟩

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 7

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

  1. ⟨6,1⟩\langle 6, 1 \rangle⟨6,1⟩
  2. ⟨8,−3⟩\langle 8, -3 \rangle⟨8,−3⟩
  3. ⟨−2,5⟩\langle -2, 5 \rangle⟨−2,5⟩
  4. ⟨2,−5⟩\langle 2, -5 \rangle⟨2,−5⟩ (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}m with initial point (3,−2)(3, -2)(3,−2) and terminal point (8,−5)(8, -5)(8,−5): m⃗=⟨8−3,−5−(−2)⟩=⟨5,−3⟩\vec{m} = \langle 8-3, -5-(-2) \rangle = \langle 5, -3 \ranglem=⟨8−3,−5−(−2)⟩=⟨5,−3⟩ For vector n⃗\vec{n}n with initial point (−1,4)(-1, 4)(−1,4) and terminal point (2,6)(2, 6)(2,6): n⃗=⟨2−(−1),6−4⟩=⟨3,2⟩\vec{n} = \langle 2-(-1), 6-4 \rangle = \langle 3, 2 \ranglen=⟨2−(−1),6−4⟩=⟨3,2⟩ Now you can find m⃗−n⃗\vec{m} - \vec{n}m−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 \ranglem−n=⟨5,−3⟩−⟨3,2⟩=⟨5−3,−3−2⟩=⟨2,−5⟩ This confirms answer choice D is correct. Looking at the wrong answers: Choice A, ⟨6,1⟩\langle 6, 1 \rangle⟨6,1⟩, likely comes from adding the vectors instead of subtracting. Choice B, ⟨8,−3⟩\langle 8, -3 \rangle⟨8,−3⟩, appears to use just the terminal point of m⃗\vec{m}m without proper vector subtraction. Choice C, ⟨−2,5⟩\langle -2, 5 \rangle⟨−2,5⟩, results from calculating n⃗−m⃗\vec{n} - \vec{m}n−m instead of m⃗−n⃗\vec{m} - \vec{n}m−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 8

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

  1. ⟨−3,−2⟩\langle -3, -2 \rangle⟨−3,−2⟩
  2. ⟨3,−6⟩\langle 3, -6 \rangle⟨3,−6⟩ (correct answer)
  3. ⟨−3,−6⟩\langle -3, -6 \rangle⟨−3,−6⟩
  4. ⟨3,6⟩\langle 3, 6 \rangle⟨3,6⟩

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 9

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

  1. ⟨1,5⟩\langle 1,5\rangle⟨1,5⟩
  2. ⟨−5,7⟩\langle -5,7\rangle⟨−5,7⟩ (correct answer)
  3. ⟨5,−7⟩\langle 5,-7\rangle⟨5,−7⟩
  4. ⟨−6,−6⟩\langle -6,-6\rangle⟨−6,−6⟩

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 10

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

  1. v⃗−w⃗\vec v-\vec wv−w is the vector from the tip of v⃗\vec vv to the tip of w⃗\vec ww.
  2. v⃗−w⃗\vec v-\vec wv−w is the vector from the tail of v⃗\vec vv to the tail of w⃗\vec ww.
  3. v⃗−w⃗\vec v-\vec wv−w is the vector from the tip of w⃗\vec ww to the tip of v⃗\vec vv. (correct answer)
  4. v⃗−w⃗\vec v-\vec wv−w is the same as w⃗−v⃗\vec w-\vec vw−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 11

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

  1. v⃗−w⃗=w⃗−v⃗\vec v-\vec w=\vec w-\vec vv−w=w−v.
  2. v⃗−w⃗=−(w⃗−v⃗)\vec v-\vec w=-(\vec w-\vec v)v−w=−(w−v). (correct answer)
  3. v⃗−w⃗=v⃗+w⃗\vec v-\vec w=\vec v+\vec wv−w=v+w.
  4. v⃗−w⃗\vec v-\vec wv−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 12

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

  1. ⟨3,0⟩\langle 3, 0\rangle⟨3,0⟩ (correct answer)
  2. ⟨5,0⟩\langle 5, 0\rangle⟨5,0⟩
  3. ⟨−3,0⟩\langle -3, 0\rangle⟨−3,0⟩
  4. ⟨4,1⟩\langle 4, 1\rangle⟨4,1⟩

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 13

For vectors v⃗=⟨1,−3⟩\vec v = \langle 1, -3\ranglev=⟨1,−3⟩ and w⃗=⟨−4,2⟩\vec w = \langle -4, 2\ranglew=⟨−4,2⟩, use v⃗−w⃗=⟨v1−w1, v2−w2⟩\vec v-\vec w=\langle v_1-w_1,\, v_2-w_2\ranglev−w=⟨v1​−w1​,v2​−w2​⟩ (equivalently v⃗+(−w⃗)\vec v+(-\vec w)v+(−w)). What is v⃗−w⃗\vec v-\vec wv−w?

  1. ⟨−3,−1⟩\langle -3, -1\rangle⟨−3,−1⟩
  2. ⟨−5,5⟩\langle -5, 5\rangle⟨−5,5⟩
  3. ⟨5,−5⟩\langle 5, -5\rangle⟨5,−5⟩ (correct answer)
  4. ⟨−4,−6⟩\langle -4, -6\rangle⟨−4,−6⟩

Explanation: This question tests understanding of vector subtraction and how to represent it using the component-wise formula. 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 = ⟨1, -3⟩ and w = ⟨-4, 2⟩, we compute v - w = ⟨1 - (-4), -3 - 2⟩ = ⟨5, -5⟩. Choice C is correct because it applies the component subtraction formula correctly, giving ⟨5, -5⟩. Choice B reverses the subtraction order, computing w - v instead of v - w, which gives the opposite vector with all signs flipped: ⟨-5, 5⟩. 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.

Question 14

Let v⃗=⟨−1,4⟩\vec v = \langle -1, 4\ranglev=⟨−1,4⟩ and w⃗=⟨3,−2⟩\vec w = \langle 3, -2\ranglew=⟨3,−2⟩. Since v⃗−w⃗=v⃗+(−w⃗)\vec v-\vec w=\vec v+(-\vec w)v−w=v+(−w) and −w⃗=⟨−w1,−w2⟩-\vec w=\langle -w_1, -w_2\rangle−w=⟨−w1​,−w2​⟩, what is v⃗−w⃗\vec v-\vec wv−w in component form?

  1. ⟨−4,6⟩\langle -4, 6\rangle⟨−4,6⟩ (correct answer)
  2. ⟨2,2⟩\langle 2, 2\rangle⟨2,2⟩
  3. ⟨−3,−8⟩\langle -3, -8\rangle⟨−3,−8⟩
  4. ⟨−4,2⟩\langle -4, 2\rangle⟨−4,2⟩

Explanation: This question tests understanding of vector subtraction and how to represent it as addition of the opposite. 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 = ⟨-1, 4⟩ and w = ⟨3, -2⟩, we compute v - w = ⟨-1 - 3, 4 - (-2)⟩ = ⟨-4, 6⟩. Choice A is correct because it applies the component subtraction formula correctly, giving ⟨-4, 6⟩. Choice D makes an arithmetic error in the subtraction, calculating ⟨-4, 2⟩ instead of ⟨-4, 6⟩ by incorrectly computing 4 - (-2) = 2 rather than 6. 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 15

Given vectors x⃗=⟨−2,6⟩\vec{x} = \langle -2, 6 \ranglex=⟨−2,6⟩ and y⃗=⟨4,−1⟩\vec{y} = \langle 4, -1 \rangley​=⟨4,−1⟩, if z⃗=2x⃗−3y⃗\vec{z} = 2\vec{x} - 3\vec{y}z=2x−3y​, what are the components of z⃗\vec{z}z?

  1. ⟨8,9⟩\langle 8, 9 \rangle⟨8,9⟩
  2. ⟨−10,9⟩\langle -10, 9 \rangle⟨−10,9⟩
  3. ⟨−16,15⟩\langle -16, 15 \rangle⟨−16,15⟩ (correct answer)
  4. ⟨−4,18⟩\langle -4, 18 \rangle⟨−4,18⟩

Explanation: When you encounter vector operations like scalar multiplication and vector addition/subtraction, you need to apply these operations component-wise to each coordinate. To find z⃗=2x⃗−3y⃗\vec{z} = 2\vec{x} - 3\vec{y}z=2x−3y​, first calculate each scalar multiplication separately, then subtract the results. Start with 2x⃗=2⟨−2,6⟩=⟨2(−2),2(6)⟩=⟨−4,12⟩2\vec{x} = 2\langle -2, 6 \rangle = \langle 2(-2), 2(6) \rangle = \langle -4, 12 \rangle2x=2⟨−2,6⟩=⟨2(−2),2(6)⟩=⟨−4,12⟩. Next, find 3y⃗=3⟨4,−1⟩=⟨3(4),3(−1)⟩=⟨12,−3⟩3\vec{y} = 3\langle 4, -1 \rangle = \langle 3(4), 3(-1) \rangle = \langle 12, -3 \rangle3y​=3⟨4,−1⟩=⟨3(4),3(−1)⟩=⟨12,−3⟩. Now subtract: z⃗=⟨−4,12⟩−⟨12,−3⟩=⟨−4−12,12−(−3)⟩=⟨−16,15⟩\vec{z} = \langle -4, 12 \rangle - \langle 12, -3 \rangle = \langle -4 - 12, 12 - (-3) \rangle = \langle -16, 15 \ranglez=⟨−4,12⟩−⟨12,−3⟩=⟨−4−12,12−(−3)⟩=⟨−16,15⟩. This confirms answer C is correct. Let's examine why the other answers are wrong. Answer A ⟨8,9⟩\langle 8, 9 \rangle⟨8,9⟩ likely comes from incorrectly adding instead of subtracting, or mixing up the scalar multipliers. Answer B ⟨−10,9⟩\langle -10, 9 \rangle⟨−10,9⟩ suggests a calculation error in the first component, possibly from −4−6=−10-4 - 6 = -10−4−6=−10 instead of the correct −4−12=−16-4 - 12 = -16−4−12=−16. Answer D ⟨−4,18⟩\langle -4, 18 \rangle⟨−4,18⟩ appears to result from forgetting to multiply y⃗\vec{y}y​ by 3 before subtracting, giving ⟨−4,12⟩−⟨4,−1⟩=⟨−8,13⟩\langle -4, 12 \rangle - \langle 4, -1 \rangle = \langle -8, 13 \rangle⟨−4,12⟩−⟨4,−1⟩=⟨−8,13⟩, which still doesn't match. Remember: always perform scalar multiplication first, then handle vector addition or subtraction component by component. Double-check your arithmetic, especially with negative numbers, as sign errors are common in these problems.

Question 16

Let v⃗=⟨2,−4⟩\vec v = \langle 2, -4\ranglev=⟨2,−4⟩ and w⃗=⟨−3,1⟩\vec w = \langle -3, 1\ranglew=⟨−3,1⟩. Which statement correctly compares v⃗−w⃗\vec v-\vec wv−w and w⃗−v⃗\vec w-\vec vw−v? (Use the idea that subtracting is adding the opposite.)

  1. v⃗−w⃗=w⃗−v⃗\vec v-\vec w = \vec w-\vec vv−w=w−v
  2. v⃗−w⃗=− (w⃗−v⃗)\vec v-\vec w = -\,(\vec w-\vec v)v−w=−(w−v) (correct answer)
  3. v⃗−w⃗=v⃗+w⃗\vec v-\vec w = \vec v+\vec wv−w=v+w
  4. v⃗−w⃗=w⃗+v⃗\vec v-\vec w = \vec w+\vec vv−w=w+v

Explanation: This question tests understanding of the non-commutative nature of vector subtraction. 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 = ⟨2 - (-3), -4 - 1⟩ = ⟨5, -5⟩ and w - v = ⟨-3 - 2, 1 - (-4)⟩ = ⟨-5, 5⟩, 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 v - w = -(w - v). 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.

Question 17

For vectors v⃗=⟨6,−1⟩\vec v = \langle 6, -1\ranglev=⟨6,−1⟩ and w⃗=⟨−2,3⟩\vec w = \langle -2, 3\ranglew=⟨−2,3⟩, what are the components of −w⃗-\vec w−w (the opposite vector used in v⃗−w⃗=v⃗+(−w⃗)\vec v-\vec w=\vec v+(-\vec w)v−w=v+(−w))?

  1. ⟨−2,−3⟩\langle -2, -3\rangle⟨−2,−3⟩
  2. ⟨2,−3⟩\langle 2, -3\rangle⟨2,−3⟩ (correct answer)
  3. ⟨2,3⟩\langle 2, 3\rangle⟨2,3⟩
  4. ⟨−6,1⟩\langle -6, 1\rangle⟨−6,1⟩

Explanation: This question tests understanding of vector subtraction and how to represent it as addition of the opposite, specifically finding the opposite vector. If w = ⟨w₁, w₂⟩, then -w = ⟨-w₁, -w₂⟩, which means we negate both components; this gives a vector with the same length but pointing in the exact opposite direction. For w = ⟨-2, 3⟩, we compute -w = ⟨-(-2), -3⟩ = ⟨2, -3⟩. Choice B is correct because it properly negates both components for -w, giving ⟨2, -3⟩. Choice A 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).

Question 18

Suppose v⃗\vec vv and w⃗\vec ww are drawn tail-to-tail from the same point. Using the tip-to-tip method for vector subtraction, how is v⃗−w⃗\vec v-\vec wv−w represented graphically?

  1. The vector from the tip of v⃗\vec vv to the tip of w⃗\vec ww.
  2. The vector from the tail of v⃗\vec vv to the tail of w⃗\vec ww.
  3. The vector from the tip of w⃗\vec ww to the tip of v⃗\vec vv. (correct answer)
  4. The vector from the tip of v⃗\vec vv to the tail of w⃗\vec ww.

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. To find v - w graphically, we can rewrite it as v + (-w): first draw v, then draw -w (the opposite of w), place the tail of -w at the head of v, and the resultant from the tail of v to the head of -w is v - w. Choice C is correct because it properly describes the tip-to-tip method, going from the tip of w to the tip of v. 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).

Question 19

For vectors v⃗=⟨2,1⟩\vec v=\langle 2,1\ranglev=⟨2,1⟩ and w⃗=⟨5,3⟩\vec w=\langle 5,3\ranglew=⟨5,3⟩, what are the components of v⃗−w⃗\vec v-\vec wv−w? (Remember v⃗−w⃗=v⃗+(−w⃗)\vec v-\vec w=\vec v+(-\vec w)v−w=v+(−w).)

  1. ⟨−3,−2⟩\langle -3,-2\rangle⟨−3,−2⟩ (correct answer)
  2. ⟨7,4⟩\langle 7,4\rangle⟨7,4⟩
  3. ⟨3,2⟩\langle 3,2\rangle⟨3,2⟩
  4. ⟨−7,−4⟩\langle -7,-4\rangle⟨−7,−4⟩

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 A is correct because it applies the component subtraction formula correctly. Choice B adds the vectors instead of subtracting them, calculating v + w = ⟨2 + 5, 1 + 3⟩ = ⟨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 20

Two vectors a⃗=⟨4,−3⟩\vec{a} = \langle 4, -3 \ranglea=⟨4,−3⟩ and b⃗=⟨−1,5⟩\vec{b} = \langle -1, 5 \rangleb=⟨−1,5⟩ are given. A student claims that a⃗−b⃗\vec{a} - \vec{b}a−b has the same magnitude as b⃗−a⃗\vec{b} - \vec{a}b−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}a−b and b⃗−a⃗\vec{b} - \vec{a}b−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 \ranglea−b=⟨5,−8⟩ and b⃗−a⃗=⟨−5,8⟩\vec{b} - \vec{a} = \langle -5, 8 \rangleb−a=⟨−5,8⟩. These are additive inverses (opposite vectors) and therefore have the same magnitude: 25+64=89\sqrt{25 + 64} = \sqrt{89}25+64​=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.