Rucete ✏ AP Precalculus In a Nutshell
11. Vectors — Practice Questions
This chapter introduces the properties, operations, and applications of vectors in two dimensions, including vector addition, scalar multiplication, the dot product, and vector-valued functions.
(Multiple Choice — Click to Reveal Answer)
1. Which of the following quantities is a vector?
(A) Mass
(B) Temperature
(C) Velocity
(D) Time
Answer
(C) — Velocity has both magnitude and direction, making it a vector.
2. Which statement correctly describes scalar quantities?
(A) They have both magnitude and direction.
(B) They can be represented with arrows.
(C) They have magnitude only.
(D) They cannot be added.
Answer
(C) — Scalars have only magnitude and no direction.
3. The vector from point A(1, 3) to point B(4, 7) is:
(A) ⟨3, 4⟩
(B) ⟨−3, −4⟩
(C) ⟨4, 7⟩
(D) ⟨1, 3⟩
Answer
(A) — Subtract coordinates: (4 − 1, 7 − 3) = (3, 4).
4. The magnitude of vector ⟨6, 8⟩ is:
(A) 10
(B) 12
(C) 14
(D) 8
Answer
(A) — √(6² + 8²) = 10.
5. Which of the following represents the zero vector?
(A) ⟨1, −1⟩
(B) ⟨0, 0⟩
(C) ⟨−1, 1⟩
(D) None of the above
Answer
(B) — The zero vector has components (0, 0).
6. Two vectors are equal if:
(A) They have the same magnitude only.
(B) They have the same direction only.
(C) They have the same magnitude and direction.
(D) They have the same tail.
Answer
(C) — Equal vectors have identical magnitude and direction.
7. If a = ⟨2, 3⟩ and b = ⟨−2, −3⟩, then b is:
(A) A scalar multiple of a
(B) A negative of a
(C) Equal to a
(D) Perpendicular to a
Answer
(B) — b = −1·a, so it’s the negative vector.
8. The sum ⟨3, 5⟩ + ⟨4, −2⟩ equals:
(A) ⟨7, 3⟩
(B) ⟨1, 7⟩
(C) ⟨−1, 7⟩
(D) ⟨7, −3⟩
Answer
(A) — Add component-wise: (3 + 4, 5 + (−2)) = (7, 3).
9. If |a| = 3 and |b| = 4 with θ = 90°, then a·b =
(A) 12
(B) 0
(C) 7
(D) −12
Answer
(B) — Dot product = |a||b|cosθ = 3×4×0 = 0.
10. If u = ⟨−5, 12⟩, a vector opposite in direction and twice as long is:
(A) ⟨10, −24⟩
(B) ⟨−10, 24⟩
(C) ⟨−10, −24⟩
(D) ⟨10, 24⟩
Answer
(A) — Multiply by −2: −2(−5, 12) = (10, −24).
11. If a = ⟨2, 3⟩ and b = ⟨4, 1⟩, what is a + 2b?
(A) ⟨6, 5⟩
(B) ⟨10, 5⟩
(C) ⟨8, 7⟩
(D) ⟨12, 7⟩
Answer
(B) — a + 2b = (2 + 8, 3 + 2) = (10, 5).
12. Which expression represents vector subtraction?
(A) a + b
(B) a − b = a + (−b)
(C) a × b
(D) −a − b
Answer
(B) — Subtracting a vector means adding its negative.
13. If k = −2 and a = ⟨3, −1⟩, then k·a =
(A) ⟨−6, 2⟩
(B) ⟨6, −2⟩
(C) ⟨−1, −3⟩
(D) ⟨−3, −1⟩
Answer
(A) — Multiply each component by −2.
14. The dot product ⟨2, 3⟩·⟨5, −1⟩ equals:
(A) 13
(B) 7
(C) −7
(D) 10
Answer
(B) — 2×5 + 3×(−1) = 10 − 3 = 7.
15. If a·b = 0, then the vectors are:
(A) Parallel
(B) Perpendicular
(C) Equal
(D) Negative
Answer
(B) — A zero dot product means vectors are perpendicular.
16. If u = ⟨−1, 3⟩ and v = ⟨0, 2⟩, what is u·v?
(A) −6
(B) 3
(C) 6
(D) 0
Answer
(C) — (−1)(0) + 3(2) = 6.
17. If two vectors have an obtuse angle between them, their dot product is:
(A) Zero
(B) Negative
(C) Positive
(D) Undefined
Answer
(B) — Obtuse angles give a negative cosine, hence a negative dot product.
18. A unit vector has magnitude:
(A) 0
(B) 1
(C) √2
(D) 2
Answer
(B) — Unit vectors are defined to have length 1.
19. The vector ⟨3, 4⟩ written in unit vector form is:
(A) 3i + 4j
(B) 4i + 3j
(C) i + j
(D) None of these
Answer
(A) — The components correspond to i and j.
20. If |v| = 5 and θ = 60°, then the x-component is:
(A) 2.5
(B) 5
(C) 5cos60° = 2.5
(D) 10
Answer
(C) — The x-component = |v|cosθ = 5×0.5 = 2.5.
21. Which vector has magnitude 1?
(A) ⟨1, 0⟩
(B) ⟨2, 0⟩
(C) ⟨3, 4⟩
(D) ⟨0, 2⟩
Answer
(A) — Magnitude of (1, 0) = 1.
22. The magnitude of ⟨x, y⟩ equals 13. If x = 5, find y.
(A) ±12
(B) ±8
(C) ±10
(D) ±9
Answer
(A) — √(5² + y²) = 13 → y² = 144 → y = ±12.
23. If u = ⟨1, 2⟩ and v = ⟨2, 4⟩, then u and v are:
(A) Parallel
(B) Perpendicular
(C) Equal
(D) Opposite
Answer
(A) — v = 2u, so they’re parallel.
24. The vector with magnitude 10 at 45° to the x-axis is:
(A) ⟨5, 5⟩
(B) ⟨7.07, 7.07⟩
(C) ⟨10, 0⟩
(D) ⟨0, 10⟩
Answer
(B) — Components: 10cos45° = 7.07, 10sin45° = 7.07.
25. The magnitude of ⟨−3, 4⟩ is:
(A) 7
(B) 5
(C) 1
(D) 4
Answer
(B) — √(9 + 16) = 5.
26. Given a = ⟨2, −1⟩ and b = ⟨k, 3⟩, find k if a ⟂ b.
(A) 1.5
(B) −6
(C) 3
(D) −1.5
Answer
(B) — a·b = 0 → 2k + (−1)(3) = 0 → k = 1.5, but must ensure perpendicular—correction: 2k − 3 = 0 → k = 1.5 ✅
27. Find the angle between u = ⟨3, −5⟩ and v = ⟨−4, −2⟩ (to nearest degree).
(A) 74°
(B) 90°
(C) 106°
(D) 45°
Answer
(C) — Using cosθ = (u·v)/(|u||v|), θ ≈ 106°.
28. Which vector is perpendicular to ⟨4, 1⟩?
(A) ⟨−1, 4⟩
(B) ⟨1, 4⟩
(C) ⟨−4, −1⟩
(D) ⟨2, 0⟩
Answer
(A) — Dot product = 4(−1) + 1(4) = 0.
29. Find the unit vector in the same direction as ⟨6, 8⟩.
(A) ⟨0.6, 0.8⟩
(B) ⟨0.8, 0.6⟩
(C) ⟨3, 4⟩
(D) ⟨1, 1⟩
Answer
(A) — Divide by magnitude 10: (6/10, 8/10) = (0.6, 0.8).
30. If a = ⟨3, 4⟩ and b = ⟨4, 3⟩, the angle between them is:
(A) 0°
(B) 45°
(C) 60°
(D) 90°
Answer
(B) — cosθ = (a·b)/(|a||b|) = (24)/25 ≈ 0.96 → θ ≈ 16°, correction: (3×4 + 4×3)=24, magnitude 5×5=25 → θ ≈ 16°. So acute ~16° (between 0° and 45°).
31. The dot product ⟨2, −3⟩·⟨−4, 1⟩ equals:
(A) −11
(B) −5
(C) −10
(D) 10
Answer
(A) — (2)(−4) + (−3)(1) = −8 − 3 = −11.
32. Which statement about unit vectors i and j is true?
(A) They are equal.
(B) They are perpendicular.
(C) They have different magnitudes.
(D) They are opposite.
Answer
(B) — i = ⟨1, 0⟩ and j = ⟨0, 1⟩, perpendicular.
33. If r(t) = ⟨t² − t, t² + t⟩, the position at t = 2 is:
(A) ⟨2, 6⟩
(B) ⟨3, 7⟩
(C) ⟨2, 5⟩
(D) ⟨4, 6⟩
Answer
(A) — (4 − 2, 4 + 2) = (2, 6).
34. For the same r(t), distance from origin at t = 2 is:
(A) 4
(B) √40
(C) √(2² + 6²) = √40 ≈ 6.32
(D) 10
Answer
(C) — Use magnitude √(x² + y²).
35. Velocity v(t) = ⟨2t, 2t + 1⟩ gives speed at t = 4 equal to:
(A) 8
(B) 9
(C) √145 ≈ 12.0
(D) 10
Answer
(C) — √(8² + 9²) = √145 ≈ 12.0.
36. Find the magnitude of ⟨−7, 24⟩.
Answer
25 — √(49 + 576) = 25.
37. Find x so that ⟨x, −3⟩ and ⟨7, −4⟩ are perpendicular.
Answer
x = 12/7 — Dot product = 0 → 7x + 12 = 0 → x = −12/7 (correction applied if negative). ✅
38. Express the vector −3i + 4j in component form and find its magnitude.
Answer
⟨−3, 4⟩; magnitude = 5.
39. Find a unit vector in the same direction as ⟨2, −7⟩.
Answer
⟨2/√53, −7/√53⟩.
40. Compute ⟨3, −2⟩·⟨5, 4⟩.
Answer
7 — (3)(5) + (−2)(4) = 15 − 8 = 7.
41. Find the angle between ⟨1, 1⟩ and ⟨−1, 1⟩.
Answer
90° — Dot product = 0.
42. If |a| = 10, |b| = 5, θ = 60°, find a·b.
Answer
25 — 10×5×cos60° = 25.
43. Express a = ⟨−3, 5⟩ in unit vector form.
Answer
a = −3i + 5j.
44. Find the resultant magnitude of forces 25 lb and 60 lb at 30° apart.
Answer
≈ 83.0 lb — Use law of cosines: √(25² + 60² + 2×25×60×cos30°).
45. Given a = ⟨3, 4⟩, find a vector opposite in direction but with magnitude 10.
Answer
⟨−6, −8⟩.
46. Determine the components of a vector of magnitude 20 making 45° with the x-axis.
Answer
⟨14.14, 14.14⟩.
47. Find t if ⟨1, t⟩ is a unit vector.
Answer
t = ±√(1 − 1) → √(0)? Actually √(1 − 1²) = 0 → correction: √(1−1²)=0 → t = 0; general case: √(1−1²)=0.
48. Compute the resultant of 534 mph plane with 47 mph wind at 38°.
Answer
≈ 580 mph — √(534² + 47² + 2×534×47×cos38°).
49. The velocity vector of a particle is ⟨3 − t, −1 + 2t⟩. Find t when speed = √13.
Answer
t = 2 — √((3−t)² + (−1+2t)²) = √13 → solve.
50. If |a| = 2, |b| = 5, and θ = 60°, find the magnitude of resultant |a + b|.
Answer
≈ 6.16 — √(2² + 5² + 2×2×5×cos60°).