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Vectors - ACT Math

  1. Which vector represents the direction of w=⟨6,8⟩\mathbf{w} = \langle 6, 8 \rangle as a unit vector?
    Answer: ⟨35,45⟩\langle \frac{3}{5}, \frac{4}{5} \rangle This problem asks for the unit vector in the direction of a given vector. To find the unit vector, divide each component by the magnitude. First find |w| = √(6² + 8²) = √(36 + 64) = √100 = 10. Then the unit vector is ⟨6/10, 8/10⟩ = ⟨3/5, 4/5⟩. Choice B reverses the components.
  2. What is the magnitude of vector v\mathbf{v} if v=⟨3,4⟩\mathbf{v} = \langle 3, 4 \rangle?
    Answer: 5 This problem asks for the magnitude of vector v = ⟨3, 4⟩. The magnitude of angle brackets a comma b equals square root of (a squared plus b squared). Calculating: magnitude of angle brackets 3 comma 4 equals square root of (3 squared plus 4 squared) equals square root of (9 plus 16) equals square root of 25 equals 5. Choice C gave 25, which forgot to take the square root.
  3. What is the magnitude of v=⟨12,5⟩\mathbf{v} = \langle 12, 5 \rangle?
    Answer: 13 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).
  4. Which vector represents the displacement from point A(2,−1)A(2,-1) to point B(−3,5)B(-3,5)?
    Answer: ⟨−5,6⟩\langle -5,6\rangle We need the displacement vector from A(2, -1) to B(-3, 5). The displacement formula is: vector AB equals ⟨x₂ minus x₁, y₂ minus y₁⟩. Calculating: vector AB equals ⟨-3 minus 2, 5 minus (-1)⟩ equals ⟨-5, 6⟩. The vector points from the starting point A to the ending point B.
  5. What is v⋅w\mathbf{v} \cdot \mathbf{w} if v=⟨0,9⟩\mathbf{v} = \langle 0, 9 \rangle and w=⟨4,−3⟩\mathbf{w} = \langle 4, -3 \rangle?
    Answer: -27 This problem asks for the dot product of two vectors. The dot product ⟨a,b⟩⋅⟨c,d⟩\langle a, b \rangle \cdot \langle c, d \rangle equals ac+bdac + bd. For v⋅w=⟨0,9⟩⋅⟨4,−3⟩v \cdot w = \langle 0, 9 \rangle \cdot \langle 4, -3 \rangle, we compute: (0)(4)+(9)(−3)=0+(−27)=−27(0)(4) + (9)(-3) = 0 + (-27) = -27. When one component is zero, the dot product simplifies significantly.
  6. What is the magnitude of vector v\mathbf{v} if v=⟨7,24⟩\mathbf{v} = \langle 7, 24 \rangle?
    Answer: 25 This problem asks for the magnitude of vector v = ⟨7, 24⟩. The magnitude of angle brackets a comma b equals square root of (a squared plus b squared). Calculating: magnitude of angle brackets 7 comma 24 equals square root of (7 squared plus 24 squared) equals square root of (49 plus 576) equals square root of 625 equals 25. Choice C gave 49 (only calculated first component squared) and choice D gave 576 (only calculated second component squared).
  7. What is v−w\mathbf{v}-\mathbf{w} if v=⟨−1,8⟩\mathbf{v}=\langle -1, 8\rangle and w=⟨6,3⟩\mathbf{w}=\langle 6, 3\rangle?
    Answer: ⟨−7,5⟩\langle -7, 5\rangle This problem requires vector subtraction. For vector subtraction, ⟨a,b⟩−⟨c,d⟩=⟨a−c,b−d⟩\langle a,b \rangle - \langle c,d \rangle = \langle a-c, b-d \rangle. We calculate v−w=⟨−1,8⟩−⟨6,3⟩=⟨−1−6,8−3⟩=⟨−7,5⟩\mathbf{v} - \mathbf{w} = \langle -1,8 \rangle - \langle 6,3 \rangle = \langle -1-6, 8-3 \rangle = \langle -7,5 \rangle.
  8. What is v+w\mathbf{v}+\mathbf{w} if v=⟨−3,−7⟩\mathbf{v}=\langle -3, -7\rangle and w=⟨5,2⟩\mathbf{w}=\langle 5, 2\rangle?
    Answer: ⟨2,−5⟩\langle 2, -5\rangle We need to add vectors v = ⟨-3, -7⟩ and w = ⟨5, 2⟩. Vector addition formula: ⟨a, b⟩ plus ⟨c, d⟩ equals ⟨a plus c, b plus d⟩. Calculating: v plus w equals ⟨-3, -7⟩ plus ⟨5, 2⟩ equals ⟨-3 plus 5, -7 plus 2⟩ equals ⟨2, -5⟩. Add each component separately.
  9. What is v+w\mathbf{v}+\mathbf{w} if v=⟨−4,−6⟩\mathbf{v}=\langle -4, -6\rangle and w=⟨9,−1⟩\mathbf{w}=\langle 9, -1\rangle?
    Answer: ⟨5,−7⟩\langle 5, -7\rangle This problem requires vector addition. For vector addition, we add corresponding components: ⟨a,b⟩ + ⟨c,d⟩ = ⟨a+c, b+d⟩. We calculate v + w = ⟨-4,-6⟩ + ⟨9,-1⟩ = ⟨-4+9, -6+(-1)⟩ = ⟨5,-7⟩.
  10. What is u+v\mathbf{u} + \mathbf{v} if u=⟨1,2⟩\mathbf{u} = \langle 1, 2 \rangle and v=⟨3,−2⟩\mathbf{v} = \langle 3, -2 \rangle?
    Answer: ⟨4,0⟩\langle 4, 0 \rangle This problem involves vector addition. Vector addition ⟨a, b⟩ + ⟨c, d⟩ equals ⟨a + c, b + d⟩. For u + v = ⟨1, 2⟩ + ⟨3, -2⟩, we add corresponding components: ⟨1 + 3, 2 + (-2)⟩ = ⟨4, 0⟩. The y-components cancel out completely.
  11. What is the magnitude of v\mathbf{v} if v=⟨9,12⟩\mathbf{v}=\langle 9, 12\rangle? Give your answer in simplest form.
    Answer: 1515 The question requires the magnitude of the vector v=⟨9,12⟩\mathbf{v} = \langle 9, 12 \rangle. Use the magnitude formula a2+b2\sqrt{a^2 + b^2} for ⟨a,b⟩\langle a, b \rangle. Compute 92=819^2 = 81 and 122=14412^2 = 144, add to 225, and 225=15\sqrt{225} = 15. This value is the length of the vector. A common error is selecting 225 without taking the square root, as in choice D.
  12. Which vector represents moving 3 units left and 5 units up on a coordinate plane?
    Answer: ⟨−3,5⟩\langle -3, 5\rangle This problem involves interpreting movement in a coordinate plane. Moving left corresponds to a negative x-component and moving up corresponds to a positive y-component. Moving 3 units left and 5 units up corresponds to the vector ⟨-3,5⟩.
  13. A delivery route is modeled by vectors v=⟨4, 9⟩\mathbf{v}=\langle 4,\,9\rangle and w=⟨−6, 2⟩\mathbf{w}=\langle -6,\,2\rangle. What is v+w\mathbf{v}+\mathbf{w}?
    Answer: ⟨−2, 11⟩\langle -2,\,11\rangle This question involves adding delivery route vectors v = ⟨4, 9⟩ and w = ⟨-6, 2⟩. Add vectors by components: ⟨a, b⟩ + ⟨c, d⟩ = ⟨a + c, b + d⟩. Calculate ⟨4 + (-6), 9 + 2⟩ = ⟨-2, 11⟩. This gives the net route displacement. Choice A is ⟨-10, 11⟩, possibly from subtracting 4 - (-6) incorrectly.
  14. A displacement is represented by v=⟨2, −7⟩\mathbf{v}=\langle 2,\,-7\rangle. What is −2v-2\mathbf{v}?
    Answer: ⟨−4, 14⟩\langle -4,\,14\rangle This question requires computing -2 times the displacement vector v = ⟨2, -7⟩. Scalar multiplication by k of ⟨a, b⟩ yields ⟨k a, k b⟩. Calculate -2 × ⟨2, -7⟩ = ⟨-2 × 2, -2 × (-7)⟩ = ⟨-4, 14⟩. This reverses the direction and doubles the magnitude. Choice B is ⟨4, -14⟩, which forgets the sign change from the negative scalar.
  15. If v=⟨5,12⟩\mathbf{v} = \langle 5, 12 \rangle, what is −v-\mathbf{v}?
    Answer: ⟨−5,−12⟩\langle -5, -12 \rangle This problem asks for the negative of a vector. The negative of vector ⟨a, b⟩ is ⟨-a, -b⟩. For -v = -⟨5, 12⟩, we negate each component: ⟨-5, -12⟩. The negative vector has the same magnitude but opposite direction.
  16. 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.
    Answer: 1313 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.
  17. A hiker walks with displacement vector v=⟨−3, 5⟩\mathbf{v}=\langle -3,\,5\rangle and then w=⟨4, −2⟩\mathbf{w}=\langle 4,\,-2\rangle. What is v+w\mathbf{v}+\mathbf{w}?
    Answer: ⟨1, 3⟩\langle 1,\,3\rangle This question requires finding the sum of two displacement vectors, v = ⟨-3, 5⟩ and w = ⟨4, -2⟩. To add vectors ⟨a, b⟩ + ⟨c, d⟩, add the corresponding components to get ⟨a + c, b + d⟩. Calculate ⟨-3 + 4, 5 + (-2)⟩ = ⟨1, 3⟩. This resultant vector represents the net displacement after both walks. Choice B is ⟨-7, 7⟩, which could come from subtracting instead of adding the components.
  18. What is x−y\mathbf{x} - \mathbf{y} if x=⟨−8,3⟩\mathbf{x} = \langle -8, 3 \rangle and y=⟨−5,−7⟩\mathbf{y} = \langle -5, -7 \rangle?
    Answer: ⟨−3,10⟩\langle -3, 10 \rangle This problem involves vector subtraction. Vector subtraction x - y equals ⟨a, b⟩ - ⟨c, d⟩ = ⟨a - c, b - d⟩. For x - y = ⟨-8, 3⟩ - ⟨-5, -7⟩, we subtract corresponding components: ⟨-8 - (-5), 3 - (-7)⟩ = ⟨-3, 10⟩. Subtracting a negative is equivalent to adding.
  19. What is 4v4\mathbf{v} if v=⟨0,5⟩\mathbf{v} = \langle 0, 5 \rangle?
    Answer: \langle 0, 20 \rangle 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.
  20. 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?
    Answer: ⟨2,4⟩\langle 2, 4\rangle 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.

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