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ACT Math

ACT Math Help: Vectors

Review real example questions for Vectors in ACT Math.

Question 1

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

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

Question 2

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

  1. 13
  2. 17
  3. 25
  4. 18
Explanation: This problem asks for the magnitude of vector v = ⟨5, 12⟩. The magnitude of angle brackets a comma b equals square root of (a squared plus b squared). Calculating: magnitude of angle brackets 5 comma 12 equals square root of (5 squared plus 12 squared) equals square root of (25 plus 144) equals square root of 169 equals 13. Choice C gave 25 (forgot to add 144 before taking square root).

Question 3

What is the magnitude of the vector ⟨3,4⟩\langle 3, 4\rangle⟨3,4⟩?

  1. 777
  2. 555
  3. 7\sqrt{7}7​
  4. 252525
Explanation: We need to find the magnitude of vector ⟨3, 4⟩. The magnitude formula is: magnitude of ⟨a, b⟩ equals square root of (a squared plus b squared). Calculating: magnitude equals square root of (3 squared plus 4 squared) equals square root of (9 plus 16) equals square root of 25 equals 5. Choice D shows 25, which is the value before taking the square root.

Question 4

What is v+w\mathbf{v}+\mathbf{w}v+w if v=⟨−2,5⟩\mathbf{v}=\langle -2, 5\ranglev=⟨−2,5⟩ and w=⟨6,−1⟩\mathbf{w}=\langle 6, -1\ranglew=⟨6,−1⟩?

  1. ⟨4,4⟩\langle 4, 4\rangle⟨4,4⟩
  2. ⟨8,−6⟩\langle 8, -6\rangle⟨8,−6⟩
  3. ⟨−8,6⟩\langle -8, 6\rangle⟨−8,6⟩
  4. ⟨4,−6⟩\langle 4, -6\rangle⟨4,−6⟩
Explanation: We need to add vectors v = ⟨-2, 5⟩ and w = ⟨6, -1⟩. Vector addition formula: ⟨a, b⟩ plus ⟨c, d⟩ equals ⟨a plus c, b plus d⟩. Calculating: v plus w equals ⟨-2, 5⟩ plus ⟨6, -1⟩ equals ⟨-2 plus 6, 5 plus (-1)⟩ equals ⟨4, 4⟩. Add corresponding components separately.

Question 5

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

  1. \langle 6, -6 \rangle
  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 6

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

  1. ⟨1,4⟩\langle 1, 4 \rangle⟨1,4⟩
  2. ⟨2,6⟩\langle 2, 6 \rangle⟨2,6⟩
  3. ⟨3,6⟩\langle 3, 6 \rangle⟨3,6⟩
  4. ⟨4,6⟩\langle 4, 6 \rangle⟨4,6⟩
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 7

What is the magnitude of c=⟨8,15⟩\mathbf{c} = \langle 8, 15 \ranglec=⟨8,15⟩?

  1. 17
  2. 23
  3. 64
  4. 225
Explanation: This problem asks for the magnitude of a vector. The magnitude of vector ⟨a,b⟩\langle a, b \rangle⟨a,b⟩ equals a2+b2\sqrt{a^2 + b^2}a2+b2​. For c = ⟨8,15⟩\langle 8, 15 \rangle⟨8,15⟩, the magnitude equals 82+152=64+225=289=17\sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 1782+152​=64+225​=289​=17. This is a Pythagorean triple (8,15,17)(8, 15, 17)(8,15,17).

Question 8

What is the magnitude of the vector ⟨8,6⟩\langle 8, 6\rangle⟨8,6⟩?

  1. 141414
  2. 14\sqrt{14}14​
  3. 101010
  4. 100100100
Explanation: We need to find the magnitude of vector ⟨8, 6⟩. Using the magnitude formula: magnitude of ⟨a, b⟩ equals square root of (a squared plus b squared). Calculating: magnitude equals square root of (8 squared plus 6 squared) equals square root of (64 plus 36) equals square root of 100 equals 10. This is a 6-8-10 Pythagorean triple (scaled version of 3-4-5).

Question 9

What is −3v-3\mathbf{v}−3v if v=⟨2,−4⟩\mathbf{v}=\langle 2, -4\ranglev=⟨2,−4⟩?

  1. ⟨−6,12⟩\langle -6, 12\rangle⟨−6,12⟩
  2. ⟨6,12⟩\langle 6, 12\rangle⟨6,12⟩
  3. ⟨−6,−12⟩\langle -6, -12\rangle⟨−6,−12⟩
  4. ⟨2,12⟩\langle 2, 12\rangle⟨2,12⟩
Explanation: We need to find -3v where v = ⟨2, -4⟩. Scalar multiplication formula: k times ⟨a, b⟩ equals ⟨ka, kb⟩. Calculating: -3 times ⟨2, -4⟩ equals ⟨-3 times 2, -3 times (-4)⟩ equals ⟨-6, 12⟩. Note that -3 times (-4) gives positive 12.

Question 10

What is 4v4\mathbf{v}4v if v=⟨0,5⟩\mathbf{v} = \langle 0, 5 \ranglev=⟨0,5⟩?

  1. \langle 0, 5 \rangle
  2. \langle 4, 20 \rangle
  3. \langle 0, 20 \rangle
  4. \langle 0, 25 \rangle
Explanation: This problem asks for scalar multiplication 4v where v = ⟨0, 5⟩. For scalar multiplication, k times angle brackets a comma b equals angle brackets ka comma kb. Calculating: 4 times angle brackets 0 comma 5 equals angle brackets 4 times 0 comma 4 times 5 equals angle brackets 0 comma 20. Multiply each component by the scalar 4.