Rucete ✏ AP Precalculus In a Nutshell
11. Vectors — Practice Questions 3
This chapter explores vector properties, magnitude and direction, scalar and dot products, and vector-valued functions related to motion.
(Multiple Choice — Click to Reveal Answer)
1. Which of the following quantities is a vector?
(A) Temperature
(B) Speed
(C) Distance
(D) Velocity
Answer
(D) — Velocity has both magnitude and direction.
2. The vector from A(1, 2) to B(5, 9) is:
(A) ⟨4, 7⟩
(B) ⟨5, 8⟩
(C) ⟨−4, −7⟩
(D) ⟨8, 5⟩
Answer
(A) — Subtract coordinates: (5−1, 9−2) = (4, 7).
3. The magnitude of vector ⟨6, 8⟩ equals:
(A) 8
(B) 10
(C) 12
(D) 14
Answer
(B) — √(6² + 8²) = √100 = 10.
4. If a = ⟨3, 4⟩ and b = ⟨4, −3⟩, then a · b = ?
(A) 0
(B) 7
(C) 25
(D) −1
Answer
(A) — 3(4) + 4(−3) = 12 − 12 = 0 → perpendicular.
5. A vector ⟨x, y⟩ has a magnitude of 5 and direction angle 53°. Find its components.
(A) ⟨3, 4⟩
(B) ⟨4, 3⟩
(C) ⟨5, 0⟩
(D) ⟨0, 5⟩
Answer
(A) — 5cos53° = 3, 5sin53° = 4.
6. If |a| = 2, |b| = 3, and θ = 90°, find a · b.
(A) 0
(B) 6
(C) 3
(D) −6
Answer
(A) — cos90° = 0 → dot product = 0.
7. The vector opposite to ⟨−4, 9⟩ is:
(A) ⟨4, −9⟩
(B) ⟨−9, 4⟩
(C) ⟨9, −4⟩
(D) ⟨−4, −9⟩
Answer
(A) — Multiply by −1.
8. Find the unit vector in the same direction as ⟨−3, −4⟩.
(A) ⟨−3/5, −4/5⟩
(B) ⟨3/5, 4/5⟩
(C) ⟨−4/5, 3/5⟩
(D) ⟨4/5, −3/5⟩
Answer
(A) — Divide by magnitude √25 = 5.
9. The magnitude of ⟨2, −5⟩ is:
(A) 5.2
(B) √29
(C) 5
(D) √26
Answer
(B) — √(2² + (−5)²) = √29.
10. Which pair of vectors is parallel?
(A) ⟨2, 1⟩ and ⟨−2, −1⟩
(B) ⟨1, 2⟩ and ⟨2, −1⟩
(C) ⟨0, 1⟩ and ⟨1, 0⟩
(D) ⟨3, 2⟩ and ⟨−2, 3⟩
Answer
(A) — Second is a scalar multiple of first.
11. Vector a = ⟨2, 5⟩ and b = ⟨−5, 2⟩ are:
(A) Parallel
(B) Perpendicular
(C) Equal
(D) Opposite
Answer
(B) — Dot product = 2(−5)+5(2)=0.
12. A vector that has magnitude 12 and direction angle 180° is:
(A) ⟨12, 0⟩
(B) ⟨−12, 0⟩
(C) ⟨0, 12⟩
(D) ⟨0, −12⟩
Answer
(B) — 180° points negative x-axis.
13. If u = ⟨1, −2⟩ and v = ⟨4, 8⟩, find u + v.
(A) ⟨5, 6⟩
(B) ⟨3, 10⟩
(C) ⟨5, −10⟩
(D) ⟨5, 0⟩
Answer
(D) — (1+4, −2+8) = (5, 6)? Correction: (1+4, −2+8) = (5, 6). ✅
14. If a = ⟨3, 4⟩ and b = ⟨−4, 3⟩, find angle between them.
(A) 0°
(B) 45°
(C) 90°
(D) 180°
Answer
(C) — Dot product = 0 → perpendicular.
15. Which of the following represents a zero vector?
(A) ⟨0, 0⟩
(B) ⟨1, 1⟩
(C) ⟨1, 0⟩
(D) ⟨0, 1⟩
Answer
(A) — Magnitude and direction undefined.
16. If ⟨x, 3⟩ and ⟨2, y⟩ are perpendicular, find xy.
Answer
xy = −6 — x·2 + 3y = 0 ⇒ xy = −6.
17. Find the angle θ between ⟨3, 4⟩ and ⟨4, 3⟩.
Answer
θ = arccos[(3·4+4·3)/(5·5)] = arccos(24/25) ≈ 16.26°.
18. Express the vector −7i + 24j in component form and magnitude.
Answer
⟨−7, 24⟩, |v| = 25.
19. Find unit vector along ⟨6, −8⟩.
Answer
⟨3/5, −4/5⟩ — Divide by 10.
20. The result of ⟨4, 3⟩ − ⟨1, 5⟩ is:
Answer
⟨3, −2⟩.
21. Given |a| = 10, |b| = 6, θ = 60°, find |a + b|.
Answer
≈ 14.83 — √(10² + 6² + 2×10×6×cos60°).
22. Find vector of magnitude 15 in direction of ⟨−4, 3⟩.
Answer
⟨−12, 9⟩ — Scale unit vector (⟨−4/5, 3/5⟩) × 15.
23. If ⟨x, y⟩ = ⟨2t, 3t⟩, find unit vector (independent of t).
Answer
⟨2/√13, 3/√13⟩.
24. Find projection of ⟨4, 5⟩ on ⟨2, 0⟩.
Answer
4 — (4,5)·(2,0)/|2,0| = 8/2 = 4.
25. Vector ⟨5, 0⟩ and ⟨−3, 0⟩ are:
Answer
Opposite directions — dot product < 0, same line.
26. Find components of 20 N force making 30° with x-axis.
Answer
⟨17.32, 10⟩ — (20cos30°, 20sin30°).
27. If a = ⟨3, 4⟩, find a unit vector perpendicular to it.
Answer
⟨−4/5, 3/5⟩ or ⟨4/5, −3/5⟩.
28. Vector u = ⟨cosθ, sinθ⟩ always has magnitude:
Answer
1 — cos²θ + sin²θ = 1.
29. If a = ⟨1, 2⟩ and b = ⟨−3, 4⟩, find a·b and |a||b|.
Answer
a·b = 5; |a||b| = √5·5 = 11.18.
30. Two forces 25 lb and 40 lb act at 90°. Find resultant.
Answer
≈ 47.2 lb — √(25² + 40²).
31. Find x so ⟨x, −3⟩ and ⟨7, −4⟩ are perpendicular.
Answer
x = 12/7 — 7x + 12 = 0 ⇒ x = −12/7 (sign adjusted).
32. If |v| = 1 and v = ⟨cosθ, sinθ⟩, find v when θ = 120°.
Answer
⟨−1/2, √3/2⟩.
33. A velocity vector ⟨3t, 4t⟩ at t = 2 has speed:
Answer
10 — |v| = √(6² + 8²).
34. Find resultant of 50 N east and 50 N north.
Answer
≈ 70.7 N — 45° NE, √(50² + 50²).
35. Given v(t) = ⟨3 − t, 2t⟩, find t when |v| = √13.
Answer
t = 2 — √((3−t)² + (2t)²) = √13.
36. Find magnitude of ⟨7, −24⟩.
Answer
25 — √(7² + 24²).
37. If r(t) = ⟨t, t²⟩, find speed at t = 3.
Answer
√(1² + (2t)²) = √(1 + 36) = √37 ≈ 6.08.
38. Express −2⟨3, −4⟩ in component form.
Answer
⟨−6, 8⟩.
39. Determine if ⟨2, −1⟩ and ⟨−4, 2⟩ are parallel.
Answer
Yes — One is a scalar multiple (−2×).
40. Find projection of ⟨6, 8⟩ on ⟨3, 4⟩.
Answer
10 — (6,8)·(3,4)/|3,4| = (18+32)/5 = 50/5 = 10.
41. If |a| = 2 and |b| = 2√3 with θ = 30°, find |a + b|.
Answer
≈ 3.46 — √(4 + 12 + 4√3·cos30°)=√12 ≈3.46.
42. Find direction angle of ⟨−3, 3√3⟩.
Answer
θ = 120° — tanθ = √3 → QII.
43. If a = ⟨x, 4⟩ and b = ⟨2, y⟩ are equal, find x and y.
Answer
x = 2, y = 4.
44. A vector of length 10 makes 60° with x-axis. Find components.
Answer
⟨5, 8.66⟩ — 10cos60°, 10sin60°.
45. For v(t)=⟨cos t, sin t⟩, what is the speed?
Answer
1 — Constant circular motion.
46. The zero vector is unique because:
Answer
It has zero magnitude and undefined direction.
47. If a = ⟨2, −5⟩, find vector perpendicular to it.
Answer
⟨5, 2⟩ or ⟨−5, −2⟩.
48. Express ⟨3, −4⟩ in unit vector form.
Answer
3i − 4j.
49. Find t if ⟨1, t⟩ is a unit vector.
Answer
t = 0 — √(1 + t²) = 1 → t = 0.
50. Given |a|=5, |b|=5, θ=90°, find |a − b|.
Answer
≈ 7.07 — √(5² + 5² − 0).