ACT Math Quiz: Vectors
20 questions · exam conditions
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VectorsQuestion 1 of 20

Which vector represents 4b4\mathbf{b} if b=⟨0,7⟩\mathbf{b} = \langle 0, 7 \rangle?

⟨0,28⟩\langle 0, 28 \rangle
⟨0,7⟩\langle 0, 7 \rangle
⟨4,7⟩\langle 4, 7 \rangle
⟨4,28⟩\langle 4, 28 \rangle
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ACT Math Quiz

ACT Math Quiz: Vectors

Practice Vectors in ACT Math 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 Vectors, giving you a quick way to practice the rules, question types, and explanations that matter most for ACT Math.

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

Which vector represents 4b4\mathbf{b} if b=⟨0,7⟩\mathbf{b} = \langle 0, 7 \rangle?

  1. ⟨0,28⟩\langle 0, 28 \rangle (correct answer)
  2. ⟨0,7⟩\langle 0, 7 \rangle
  3. ⟨4,7⟩\langle 4, 7 \rangle
  4. ⟨4,28⟩\langle 4, 28 \rangle
Explanation: This problem requires scalar multiplication of a vector. When multiplying vector ⟨a,b⟩\langle a, b \rangle by scalar k, the result is ⟨ka,kb⟩\langle ka, kb \rangle. For 4b = 4⟨0,7⟩\langle 0, 7 \rangle, we multiply each component by 4: ⟨4⋅0,4⋅7⟩\langle 4 \cdot 0, 4 \cdot 7 \rangle = ⟨0,28⟩\langle 0, 28 \rangle. Scalar multiplication affects each component independently.

Question 2

A boat's velocity is v=⟨−2,1⟩v=\langle -2,1\rangle (in m/s). Which vector represents the opposite direction with the same speed?

  1. ⟨−2,−1⟩\langle -2,-1\rangle
  2. ⟨2,1⟩\langle 2,1\rangle
  3. ⟨2,−1⟩\langle 2,-1\rangle (correct answer)
  4. ⟨−1,2⟩\langle -1,2\rangle
Explanation: The boat's velocity is v = ⟨-2, 1⟩ and we need the opposite direction with same speed. The opposite direction is found by negating the vector: -v equals -1 times ⟨-2, 1⟩ equals ⟨2, -1⟩. This reverses the direction while maintaining the same magnitude (speed). The magnitude remains square root of 5 in both cases.

Question 3

If v=⟨7,1⟩\mathbf{v} = \langle 7, 1 \rangle and w=⟨1,7⟩\mathbf{w} = \langle 1, 7 \rangle, what is v−w\mathbf{v} - \mathbf{w}?

  1. \langle 6, -6 \rangle (correct answer)
  2. \langle 8, 8 \rangle
  3. \langle 6, 6 \rangle
  4. \langle -6, 6 \rangle
Explanation: This problem asks for vector subtraction v - w where v = ⟨7, 1⟩ and w = ⟨1, 7⟩. For vector subtraction, angle brackets a comma b minus angle brackets c comma d equals angle brackets a minus c comma b minus d. Calculating: v minus w equals angle brackets 7 comma 1 minus angle brackets 1 comma 7 equals angle brackets 7 minus 1 comma 1 minus 7 equals angle brackets 6 comma negative 6. Subtract corresponding components.

Question 4

A skier's displacement changes from v=⟨2,−6⟩\mathbf{v}=\langle 2, -6\rangle to w=⟨−5,1⟩\mathbf{w}=\langle -5, 1\rangle. What is v−w\mathbf{v}-\mathbf{w}?

  1. ⟨−3,−5⟩\langle -3, -5\rangle
  2. ⟨7,−7⟩\langle 7, -7\rangle (correct answer)
  3. ⟨−7,7⟩\langle -7, 7\rangle
  4. ⟨3,5⟩\langle 3, 5\rangle
Explanation: This question requires computing v - w, where v = ⟨2, -6⟩ and w = ⟨-5, 1⟩. Vector subtraction is done by subtracting components: ⟨a, b⟩ - ⟨c, d⟩ = ⟨a - c, b - d⟩. Calculate ⟨2 - (-5), -6 - 1⟩ = ⟨7, -7⟩. This emphasizes subtracting each corresponding component to find the difference vector. The wording about displacement change might suggest w - v as ⟨-7, 7⟩, but the question explicitly asks for v - w.

Question 5

What is v+w\mathbf{v} + \mathbf{w} if v=⟨1,2⟩\mathbf{v} = \langle 1, 2 \rangle and w=⟨3,4⟩\mathbf{w} = \langle 3, 4 \rangle?

  1. ⟨1,4⟩\langle 1, 4 \rangle
  2. ⟨2,6⟩\langle 2, 6 \rangle
  3. ⟨3,6⟩\langle 3, 6 \rangle
  4. ⟨4,6⟩\langle 4, 6 \rangle (correct answer)
Explanation: This problem involves vector addition. When adding vectors ⟨a, b⟩ + ⟨c, d⟩, the result equals ⟨a + c, b + d⟩. For v + w = ⟨1, 2⟩ + ⟨3, 4⟩, we add corresponding components: ⟨1 + 3, 2 + 4⟩ = ⟨4, 6⟩. Vector addition requires adding components separately.

Question 6

If v=⟨10,−5⟩\mathbf{v} = \langle 10, -5 \rangle and w=⟨−3,2⟩\mathbf{w} = \langle -3, 2 \rangle, what is v+w\mathbf{v} + \mathbf{w}?

  1. \langle 7, -7 \rangle
  2. \langle 13, -7 \rangle
  3. \langle 13, -3 \rangle
  4. \langle 7, -3 \rangle (correct answer)
Explanation: This problem asks for vector addition v + w where v = ⟨10, -5⟩ and w = ⟨-3, 2⟩. For vector addition, angle brackets a comma b plus angle brackets c comma d equals angle brackets a plus c comma b plus d. Calculating: v plus w equals angle brackets 10 comma negative 5 plus angle brackets negative 3 comma 2 equals angle brackets 10 plus negative 3 comma negative 5 plus 2 equals angle brackets 7 comma negative 3. Add corresponding components.

Question 7

Which vector represents moving 4 units left and 2 units up?

  1. ⟨4,2⟩\langle 4, 2\rangle
  2. ⟨−4,−2⟩\langle -4, -2\rangle
  3. ⟨−4,2⟩\langle -4, 2\rangle (correct answer)
  4. ⟨2,−4⟩\langle 2, -4\rangle
Explanation: We need a vector representing 4 units left and 2 units up. Moving left means negative x-component, and moving up means positive y-component. The vector is ⟨-4, 2⟩. Choice A ⟨4, 2⟩ would represent moving right instead of left.

Question 8

What is 12v\tfrac{1}{2}\mathbf{v} if v=⟨10,−8⟩\mathbf{v}=\langle 10, -8\rangle?

  1. ⟨5,−4⟩\langle 5, -4\rangle (correct answer)
  2. ⟨5,4⟩\langle 5, 4\rangle
  3. ⟨20,−16⟩\langle 20, -16\rangle
  4. ⟨10,−4⟩\langle 10, -4\rangle
Explanation: We need to find (1/2)v where v = ⟨10, -8⟩. Scalar multiplication formula: k times ⟨a, b⟩ equals ⟨ka, kb⟩. Calculating: (1/2) times ⟨10, -8⟩ equals ⟨(1/2) times 10, (1/2) times (-8)⟩ equals ⟨5, -4⟩. Each component is halved.

Question 9

What is 3v3\mathbf{v} if v=⟨−3,−6⟩\mathbf{v}=\langle -3, -6\rangle?

  1. ⟨−6,−12⟩\langle -6, -12\rangle
  2. ⟨−9,−6⟩\langle -9, -6\rangle
  3. ⟨9,18⟩\langle 9, 18\rangle
  4. ⟨−9,−18⟩\langle -9, -18\rangle (correct answer)
Explanation: This problem involves scalar multiplication. For scalar multiplication, k times ⟨a,b⟩ equals ⟨ka,kb⟩. We calculate 3v = 3⟨-3,-6⟩ = ⟨3×(-3), 3×(-6)⟩ = ⟨-9,-18⟩.

Question 10

In the standard (x,y)(x, y) coordinate plane, u⃗=⟨2,−5⟩\vec{u} = \langle 2, -5 \rangle and v⃗=⟨−3,1⟩\vec{v} = \langle -3, 1 \rangle. What is the magnitude of the vector u⃗+v⃗\vec{u} + \vec{v}?

  1. 17\sqrt{17} (correct answer)
  2. 55
  3. 37\sqrt{37}
  4. 41\sqrt{41}
Explanation: This is a vectors question testing addition and magnitude. Choice A (√17) is correct — add the vectors: u + v = ⟨2 + (−3), −5 + 1⟩ = ⟨−1, −4⟩. Magnitude = √((−1)² + (−4)²) = √(1 + 16) = √17. Choice B (5) adds the absolute values of the components instead of using the distance formula: |−1| + |−4| = 1 + 4 = 5. This is the "taxicab" distance, not the Euclidean magnitude. Choice C (√37) results from an error in the vector addition step, possibly computing ⟨−1, −6⟩ and finding √(1 + 36) = √37. Choice D (√41) results from using the original components of u without performing the addition: √(2² + (−5)² ) = √(4 + 25) = √29... or from computing the magnitude of v: √(9 + 1) = √10. Pro tip: Vector addition is component-wise — add x-components, add y-components. Then apply the distance formula (√(x² + y²)) to the resulting vector. The magnitude is never found by adding components directly.

Question 11

What is 2u+3v2\mathbf{u} + 3\mathbf{v} if u=⟨1,1⟩\mathbf{u} = \langle 1, 1 \rangle and v=⟨2,3⟩\mathbf{v} = \langle 2, 3 \rangle?

  1. ⟨5,8⟩\langle 5, 8 \rangle
  2. ⟨7,11⟩\langle 7, 11 \rangle
  3. ⟨8,11⟩\langle 8, 11 \rangle (correct answer)
  4. ⟨6,9⟩\langle 6, 9 \rangle
Explanation: This problem involves linear combination of vectors. We compute 2u + 3v = 2⟨1, 1⟩ + 3⟨2, 3⟩. First multiply: 2⟨1, 1⟩ = ⟨2, 2⟩ and 3⟨2, 3⟩ = ⟨6, 9⟩. Then add: ⟨2, 2⟩ + ⟨6, 9⟩ = ⟨8, 11⟩. Apply scalar multiplication first, then vector addition.

Question 12

What is the magnitude of x=⟨9,12⟩\mathbf{x} = \langle 9, 12 \rangle?

  1. 15 (correct answer)
  2. 21
  3. 144
  4. 81
Explanation: This problem asks for the magnitude of a vector. The magnitude of vector ⟨a, b⟩ equals √(a² + b²). For x = ⟨9, 12⟩, the magnitude equals √(9² + 12²) = √(81 + 144) = √225 = 15. This is a Pythagorean triple (9, 12, 15).

Question 13

What is u⋅v\mathbf{u} \cdot \mathbf{v} if u=⟨3,−2⟩\mathbf{u} = \langle 3, -2 \rangle and v=⟨4,5⟩\mathbf{v} = \langle 4, 5 \rangle?

  1. 2 (correct answer)
  2. 7
  3. 22
  4. 23
Explanation: This problem asks for the dot product of two vectors. The dot product ⟨a, b⟩ · ⟨c, d⟩ equals ac + bd. For u · v = ⟨3, -2⟩ · ⟨4, 5⟩, we compute: (3)(4) + (-2)(5) = 12 + (-10) = 2. The dot product combines components through multiplication and addition.

Question 14

What is the magnitude of vector v=⟨3,4⟩\mathbf{v} = \langle 3, 4 \rangle?

  1. 5 (correct answer)
  2. 7
  3. 25
  4. 6
Explanation: This problem asks for the magnitude of a vector. The magnitude of a vector ⟨a, b⟩ equals the square root of (a² + b²). For vector v = ⟨3, 4⟩, the magnitude equals √(3² + 4²) = √(9 + 16) = √25 = 5. Choice C gave 25, which is the value before taking the square root.

Question 15

Which vector represents the addition of a=⟨5,−3⟩\mathbf{a} = \langle 5, -3 \rangle and b=⟨−2,4⟩\mathbf{b} = \langle -2, 4 \rangle?

  1. ⟨−7,1⟩\langle -7, 1 \rangle
  2. ⟨3,7⟩\langle 3, 7 \rangle
  3. ⟨3,1⟩\langle 3, 1 \rangle (correct answer)
  4. ⟨7,1⟩\langle 7, 1 \rangle
Explanation: This problem involves vector addition. Vector addition ⟨a, b⟩ + ⟨c, d⟩ equals ⟨a + c, b + d⟩. For a + b = ⟨5, -3⟩ + ⟨-2, 4⟩, we add corresponding components: ⟨5 + (-2), -3 + 4⟩ = ⟨3, 1⟩. Add components separately to get the sum vector.

Question 16

What is the magnitude of v=⟨12,5⟩\mathbf{v} = \langle 12, 5 \rangle?

  1. 144
  2. 17
  3. 17
  4. 13 (correct answer)
Explanation: This problem asks for the magnitude of a vector. The magnitude of vector ⟨a,b⟩\langle a, b \rangle equals √(a2+b2)√(a^2 + b^2). For v = ⟨12,5⟩\langle 12, 5 \rangle, the magnitude equals √(122+52)=√(144+25)=√169=13√(12^2 + 5^2) = √(144 + 25) = √169 = 13. This is a Pythagorean triple (5, 12, 13).

Question 17

A force vector is v=⟨4,−9⟩\mathbf{v}=\langle 4, -9\rangle. What is 3v3\mathbf{v}?

  1. ⟨12,−9⟩\langle 12, -9\rangle
  2. ⟨7,−27⟩\langle 7, -27\rangle
  3. ⟨12,−27⟩\langle 12, -27\rangle (correct answer)
  4. ⟨−12,27⟩\langle -12, 27\rangle
Explanation: This question involves scalar multiplication to find 3v, where v = ⟨4, -9⟩. Scalar multiplication by k of ⟨a, b⟩ is ⟨ka, kb⟩. Compute 3⟨4, -9⟩ = ⟨12, -27⟩. This demonstrates multiplying each component by the scalar to scale the vector. A distractor might involve adding the scalar to components, like ⟨7, -27⟩, which mixes operations incorrectly.

Question 18

A robot moves first by ⟨−4,7⟩\langle -4, 7\rangle and then by ⟨6,−3⟩\langle 6, -3\rangle. Which vector represents the robot's net movement?

  1. ⟨2,−10⟩\langle 2, -10\rangle
  2. ⟨−10,10⟩\langle -10, 10\rangle
  3. ⟨2,4⟩\langle 2, 4\rangle (correct answer)
  4. ⟨10,−10⟩\langle 10, -10\rangle
Explanation: This requires adding the robot's movement vectors <-4, 7> and <6, -3> for net movement. Add components: -4 + 6 = 2 and 7 + (-3) = 4, resulting in <2, 4>. This is the resultant displacement vector. Vector addition emphasizes combining individual movements component-wise.

Question 19

What is −2v-2\mathbf{v} if v=⟨9,−4⟩\mathbf{v}=\langle 9, -4\rangle?

  1. ⟨−18,8⟩\langle -18, 8\rangle (correct answer)
  2. ⟨18,8⟩\langle 18, 8\rangle
  3. ⟨−18,−8⟩\langle -18, -8\rangle
  4. ⟨−2,8⟩\langle -2, 8\rangle
Explanation: This problem involves scalar multiplication. For scalar multiplication, k times ⟨a,b⟩ equals ⟨ka,kb⟩. We calculate -2v = -2⟨9,-4⟩ = ⟨-2×9, -2×(-4)⟩ = ⟨-18,8⟩.

Question 20

What is 4v4\mathbf{v} if v=⟨1,−7⟩\mathbf{v}=\langle 1, -7\rangle?

  1. ⟨4,−28⟩\langle 4, -28\rangle (correct answer)
  2. ⟨4,−7⟩\langle 4, -7\rangle
  3. ⟨−4,28⟩\langle -4, 28\rangle
  4. ⟨5,−21⟩\langle 5, -21\rangle
Explanation: This problem involves scalar multiplication. For scalar multiplication, k times ⟨a,b⟩ equals ⟨ka,kb⟩. We calculate 4v = 4⟨1,-7⟩ = ⟨4×1, 4×(-7)⟩ = ⟨4,-28⟩.