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ACT Math Help: Vectors

Review real example questions for Vectors in ACT Math.

Question 1 / 10

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

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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.