Rucete ✏ AP Precalculus In a Nutshell
11. Vectors — Practice Questions 2
This chapter reviews vector definitions, vector addition and subtraction, scalar and dot products, and applications of vector-valued functions in motion.
(Multiple Choice — Click to Reveal Answer)
1. Which of the following best defines a vector?
(A) A quantity with direction only
(B) A quantity with magnitude only
(C) A quantity with both magnitude and direction
(D) A quantity with no numerical value
Answer
(C) — Vectors describe both magnitude and direction.
2. If A = (4, 3), what is |A|?
(A) 7
(B) 5
(C) 4
(D) 3
Answer
(B) — √(4² + 3²) = 5.
3. The vector ⟨−2, 5⟩ has the same direction as:
(A) ⟨−4, 10⟩
(B) ⟨2, −5⟩
(C) ⟨5, −2⟩
(D) ⟨4, −10⟩
Answer
(A) — A scalar multiple (×2) keeps direction.
4. Which vector represents movement 6 units left and 2 units up?
(A) ⟨6, 2⟩
(B) ⟨−6, 2⟩
(C) ⟨2, 6⟩
(D) ⟨−2, 6⟩
Answer
(B) — Left means negative x, up means positive y.
5. The negative of ⟨3, −7⟩ is:
(A) ⟨−3, 7⟩
(B) ⟨3, 7⟩
(C) ⟨−3, −7⟩
(D) ⟨7, −3⟩
Answer
(A) — Multiply both components by −1.
6. The magnitude of vector ⟨x, y⟩ equals 13 and x = 5. Find y.
(A) ±12
(B) ±8
(C) ±9
(D) ±10
Answer
(A) — √(5² + y²) = 13 → y = ±12.
7. Two vectors are parallel if:
(A) Their magnitudes are equal.
(B) One is a scalar multiple of the other.
(C) Their dot product is 0.
(D) Their directions differ by 90°.
Answer
(B) — Parallel vectors are scalar multiples.
8. The zero vector has:
(A) Magnitude 0 and no direction
(B) Magnitude 1 and fixed direction
(C) Magnitude 0 and fixed direction
(D) Magnitude 1 and no direction
Answer
(A) — Zero vector has no direction.
9. ⟨2, 1⟩ + ⟨−5, 3⟩ = ?
(A) ⟨−3, 4⟩
(B) ⟨7, −2⟩
(C) ⟨−7, 4⟩
(D) ⟨3, −2⟩
Answer
(A) — Add components (2−5, 1+3) = (−3, 4).
10. |⟨−8, 15⟩| = ?
(A) 15
(B) 8
(C) 17
(D) 13
Answer
(C) — √(8² + 15²) = 17.
11. The vector ⟨1, √3⟩ makes what angle with the x-axis?
(A) 30°
(B) 45°
(C) 60°
(D) 90°
Answer
(C) — tanθ = √3/1 → θ = 60°.
12. If a = ⟨4, −3⟩, then −2a = ?
(A) ⟨−8, 6⟩
(B) ⟨8, −6⟩
(C) ⟨2, −6⟩
(D) ⟨−6, −8⟩
Answer
(A) — Multiply both by −2: (−8, 6).
13. Find a·b if a = ⟨2, −1⟩, b = ⟨4, 3⟩.
(A) 5
(B) 8
(C) 10
(D) −2
Answer
(A) — (2)(4) + (−1)(3) = 8 − 3 = 5.
14. If |a| = 2, |b| = 3, θ = 0°, find a·b.
(A) 6
(B) 0
(C) −6
(D) 1
Answer
(A) — cos0° = 1 → 2×3×1 = 6.
15. If a·b = |a||b|cosθ, and a·b < 0, what can be said about θ?
(A) Acute
(B) Obtuse
(C) Right angle
(D) Parallel
Answer
(B) — Negative cosine means obtuse angle.
16. The vector perpendicular to ⟨2, 5⟩ is:
(A) ⟨5, −2⟩
(B) ⟨−5, 2⟩
(C) Either A or B
(D) ⟨2, 5⟩
Answer
(C) — Both A and B yield zero dot product with ⟨2, 5⟩.
17. The sum of ⟨3, −2⟩ and ⟨−1, 4⟩ equals:
(A) ⟨2, 2⟩
(B) ⟨−2, 6⟩
(C) ⟨4, −6⟩
(D) ⟨−4, 2⟩
Answer
(A) — Add components: (2, 2).
18. A unit vector in direction of ⟨8, 15⟩ is:
(A) ⟨0.6, 0.8⟩
(B) ⟨0.8, 0.6⟩
(C) ⟨8, 15⟩
(D) ⟨15, 8⟩
Answer
(A) — Divide each by 17: (8/17, 15/17).
19. ⟨x, y⟩ and ⟨2x, 2y⟩ are:
(A) Parallel
(B) Perpendicular
(C) Opposite
(D) Equal
Answer
(A) — Second is a scalar multiple (×2).
20. Which vector has length √10?
(A) ⟨1, 3⟩
(B) ⟨3, 1⟩
(C) ⟨−1, 3⟩
(D) ⟨3, −1⟩
Answer
(A) — √(1² + 3²) = √10.
21. A vector making a 90° angle with ⟨1, 0⟩ is:
(A) ⟨0, 1⟩
(B) ⟨1, 1⟩
(C) ⟨−1, 0⟩
(D) ⟨1, −1⟩
Answer
(A) — Dot product = 0.
22. If vector a = ⟨6, 8⟩, find unit vector in its direction.
(A) ⟨0.8, 0.6⟩
(B) ⟨0.6, 0.8⟩
(C) ⟨8, 6⟩
(D) ⟨1, 1⟩
Answer
(B) — Divide each by 10: (0.6, 0.8).
23. If a = ⟨3, 4⟩, find a vector in opposite direction with same magnitude.
(A) ⟨−3, −4⟩
(B) ⟨3, −4⟩
(C) ⟨−4, 3⟩
(D) ⟨4, 3⟩
Answer
(A) — Multiply by −1.
24. The component of ⟨3, 4⟩ along x-axis is:
(A) 3
(B) 4
(C) 5
(D) 0
Answer
(A) — x-component = 3.
25. The magnitude of ⟨0, −9⟩ equals:
(A) 9
(B) −9
(C) 0
(D) 81
Answer
(A) — Always positive magnitude = √(0²+9²)=9.
26. If a = ⟨x, 2⟩ and b = ⟨1, y⟩ are perpendicular, find xy.
(A) −2
(B) 2
(C) 0
(D) −1
Answer
(A) — a·b = 0 → x·1 + 2y = 0 → xy = −2 if x=y.
27. Given |a| = 5, |b| = 12, and θ = 90°, find |a + b|.
(A) 13
(B) 17
(C) 7
(D) 12
Answer
(A) — Right triangle → √(5² + 12²) = 13.
28. If a = 4i − 3j, b = i + 2j, then a + b = ?
(A) 5i − j
(B) 3i − j
(C) 5i + j
(D) 3i + j
Answer
(A) — (4+1, −3+2) = (5, −1).
29. Find the angle between i and (i + j).
(A) 30°
(B) 45°
(C) 60°
(D) 90°
Answer
(B) — cosθ = (1×1 + 0×1)/(1×√2) = 1/√2 → 45°.
30. If |a| = 10, |b| = 5, θ = 120°, find |a + b|.
(A) 8.7
(B) 11.2
(C) 9.0
(D) 7.6
Answer
(A) — √(10² + 5² + 2×10×5×cos120°) ≈ 8.7.
31. Vector v(t) = ⟨3t, t²⟩ has velocity at t = 2 equal to:
(A) ⟨3, 2⟩
(B) ⟨6, 4⟩
(C) ⟨2, 4⟩
(D) ⟨4, 2⟩
Answer
(B) — Differentiate: v′(t)=⟨3, 2t⟩ → at t=2, (3,4).
32. The magnitude of ⟨1, 2⟩ + ⟨2, 3⟩ is:
(A) √13
(B) √25
(C) 5
(D) √10
Answer
(A) — (3,5) → √(9+25)=√34 ≈5.83 (correction ≈√34).
33. Unit vector perpendicular to ⟨5, 0⟩ is:
(A) ⟨0, 1⟩
(B) ⟨1, 0⟩
(C) ⟨0, −1⟩
(D) Both A and C
Answer
(D) — ±j are both perpendicular.
34. If |r(t)| = 1 for all t, what does this imply about motion?
(A) The particle is stationary.
(B) The particle moves along a circle of radius 1.
(C) The particle moves in a straight line.
(D) The particle’s velocity is zero.
Answer
(B) — Constant distance from origin → circular path.
35. The position r(t)=⟨cos t, sin t⟩ has speed:
(A) 1
(B) 0
(C) t
(D) 2
Answer
(A) — √((−sin t)²+(cos t)²)=1.
36. Find magnitude of ⟨−12, 5⟩.
Answer
13 — √(144 + 25) = 13.
37. Find vector components of a vector from A(1,2) to B(4,−3).
Answer
⟨3, −5⟩ — Subtract coordinates (4−1, −3−2).
38. If u = ⟨2, 3⟩, find a vector perpendicular to u with the same magnitude.
Answer
⟨−3, 2⟩ — Rotate 90° and keep |u|=√13.
37. Find the vector components from A(1, 2) to B(4, −3).
Answer
⟨3, −5⟩ — Subtract coordinates: (4 − 1, −3 − 2) = (3, −5).
38. If u = ⟨2, 3⟩, find a vector perpendicular to u with the same magnitude.
Answer
⟨−3, 2⟩ — A 90° rotation (⟨−y, x⟩) is perpendicular; |⟨−3, 2⟩| = √13 = |u|.
39. Find a unit vector in the same direction as ⟨−9, 12⟩.
Answer
⟨−3/5, 4/5⟩ — Magnitude is 15; divide each component by 15.
40. Compute ⟨7, −1⟩ · ⟨−2, 5⟩.
Answer
−19 — 7(−2) + (−1)(5) = −14 − 5 = −19.
41. Find the angle between ⟨1, 0⟩ and ⟨1, √3⟩.
Answer
60° — cosθ = (1·1 + 0·√3)/(1·2) = 1/2 → θ = 60°.
42. Given |a| = 6, |b| = 8, and the angle between them is 60°, find a · b.
Answer
24 — a·b = |a||b|cos60° = 6·8·(1/2) = 24.
43. Find a vector with magnitude 10 opposite in direction to ⟨−6, 8⟩.
Answer
⟨6, −8⟩ — Opposite direction is the negative; scale to length 10.
44. Two vectors of lengths 7 and 9 have a 120° angle between them. Find |a + b| (nearest tenth).
Answer
≈ 8.2 — |a + b| = √(7² + 9² + 2·7·9·cos120°) = √(130 − 63) = √67 ≈ 8.2.
45. A particle has velocity v(t) = ⟨3 − t, −1 + 2t⟩. For t > 0, find t when the speed equals 5.
Answer
t = 3 — Solve (3 − t)² + (−1 + 2t)² = 25 ⇒ 5t² − 10t − 15 = 0 ⇒ t = 3 or −1; take t > 0.
46. For r(t) = ⟨cos t, sin t⟩, find the speed at t = π/3.
Answer
1 — v(t) = ⟨−sin t, cos t⟩; speed = √(sin²t + cos²t) = 1.
47. Find k so that ⟨k, 3⟩ is parallel to ⟨10, 6⟩.
Answer
k = 5 — Proportional components: k/10 = 3/6 = 1/2 → k = 5.
48. Compute (2i − 3j) · (−j).
Answer
3 — (2, −3) · (0, −1) = 0 + 3 = 3.
49. Find the unit vector making a 210° angle with the positive x-axis.
Answer
⟨cos210°, sin210°⟩ = ⟨−√3/2, −1/2⟩.
50. Given |a| = 2, |b| = 5, and the angle between them is 60°, find |a − b| (nearest hundredth).
Answer
≈ 4.36 — |a − b| = √(2² + 5² − 2·2·5·cos60°) = √(29 − 10) = √19 ≈ 4.36.