ACT Math Quiz: Vectors
20 questions · exam conditions
0:00
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
← Back to quizzes

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

A drone's horizontal displacement is represented by the vector ⟨6,8⟩\langle 6, 8\rangle. What is the magnitude of the vector ⟨6,8⟩\langle 6, 8\rangle? Give your answer in simplest form.

  1. 1414
  2. 28\sqrt{28}
  3. 1010 (correct answer)
  4. 100\sqrt{100}
Explanation: The problem requires finding the magnitude of the vector <6, 8>, which represents the drone's horizontal displacement. The magnitude of a vector <a, b> is calculated using the formula sqrt(a² + b²). To compute this, square each component: 6² = 36 and 8² = 64, then add them to get 100, and take the square root to obtain sqrt(100) = 10. This magnitude represents the straight-line distance of the displacement. A common distractor is forgetting to take the square root, which might lead to choosing 100 or another incorrect value.

Question 12

A drone's horizontal movement is represented by the vector ⟨6, 8⟩\langle 6,\,8\rangle. What is the magnitude of the vector ⟨6, 8⟩\langle 6,\,8\rangle? Give your answer in simplest form.

  1. 1414
  2. 1010 (correct answer)
  3. 100\sqrt{100}
  4. 28\sqrt{28}
Explanation: This question asks for the magnitude of the vector ⟨6, 8⟩, which represents the drone's horizontal movement. The magnitude of a vector ⟨a, b⟩ is calculated using the formula √(a² + b²). To find it, compute √(6² + 8²) = √(36 + 64) = √100 = 10. This value represents the straight-line distance of the movement. Choice A is 14, which might result from incorrectly adding 6 + 8 without squaring or taking the square root.

Question 13

What is the magnitude of v\mathbf{v} if v=⟨6,8⟩\mathbf{v}=\langle 6, 8\rangle? Give your answer in simplest form.

  1. 1414
  2. 1010 (correct answer)
  3. 100\sqrt{100}
  4. 62+82=28\sqrt{6^2+8^2}=\sqrt{28}
Explanation: This problem asks for the magnitude of a vector. The magnitude of ⟨a,b⟩\langle a,b \rangle equals a2+b2\sqrt{a^2 + b^2}. We calculate magnitude of ⟨6,8⟩\langle 6,8 \rangle = 62+82\sqrt{6^2 + 8^2} = 36+64\sqrt{36 + 64} = 100\sqrt{100} = 10. Choice C gave 100\sqrt{100} without simplifying.

Question 14

A velocity vector is v=⟨−8,6⟩\mathbf{v}=\langle -8, 6\rangle. What is the magnitude of v\mathbf{v}? Give your answer in simplest form.

  1. 28\sqrt{28}
  2. 100\sqrt{100}
  3. 1010 (correct answer)
  4. 1414
Explanation: This question seeks the magnitude of the velocity vector v = ⟨-8, 6⟩. Use the magnitude formula √(a² + b²) for ⟨a, b⟩. Compute √((-8)² + 6²) = √(64 + 36) = √100 = 10. Note that negative signs disappear when squaring, focusing on the absolute length. Distractors like √28 might come from incorrect squaring or summing, such as using 4 and 6 instead.

Question 15

A vector is ⟨−5, 12⟩\langle -5,\,12\rangle. What is the magnitude of vector ⟨−5, 12⟩\langle -5,\,12\rangle? Give your answer in simplest form.

  1. 17\sqrt{17}
  2. 169\sqrt{169}
  3. 1313 (correct answer)
  4. 1717
Explanation: This question asks for the magnitude of the vector ⟨-5, 12⟩. Use the formula √(a² + b²) for the magnitude of ⟨a, b⟩. Calculate √((-5)² + 12²) = √(25 + 144) = √169 = 13. Note that the negative sign squares to positive, so direction doesn't affect magnitude. Choice D is 17, which might come from adding | -5 | + 12 without squaring.

Question 16

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 17

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 18

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 19

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 20

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.