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Scalar Multiplication Using Components

Scalar Multiplication Using Components
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2017-02-01 15:23
VectorCoordinate GeometryTransformationGeometric ConstructionSlider Animation
In collections Vector Basics
This GeoGebra example demonstrates scalar multiplication of vectors using their components, visually showing how a vector is scaled by a scalar factor. The core mathematical concept is linear operations on vectors, specifically scalar multiplication and its effect on vector direction and magnitude. The construction uses GeoGebra commands such as Vector, Translate, Segment, Point, and Midpoint to create the original vector u, auxiliary points and segments, and a slider k to control the scaling factor. Multiple scaled copies (C', C'', etc.) are generated to illustrate the transformation. This resource is suitable for high school or early college courses in analytic geometry or linear algebra, helping students grasp the geometric meaning of vector scaling and explore dynamic variations with different coefficients.

GeoGebra Commands & Parameters

  1. Vector(A, B) → u
  2. Translate(C, Fin) → EndC
  3. Segment(C, EndC) → CEndC
  4. Translate(C, u) → C'
  5. Translate(C', u) → C''
  6. Translate(C'', u) → C'''
  7. Translate(C''', u) → C''''
  8. Segment(C, C') → a
  9. Segment(C', C'') → b
  10. Segment(C'', C''') → c
  11. Segment(C''', C'''') → d
  12. Segment(A, C) → a'
  13. Segment(A, C') → b'
  14. Segment(A, C'') → c'
  15. Segment(A, C''') → d'
  16. Point(a', n) → A_1
  17. Point(b', n - 1) → B_1
  18. Translate(A_1, u) → D
  19. Vector(A_1, D) → a_1
  20. Translate(B_1, u) → E
  21. Vector(B_1, E) → b_1
  22. Point(CEndC, n_1) → AlmEnd
  23. Vector(C, AlmEnd) → End
  24. Point(c', n - 2) → C_1
  25. Point(d', n - 3) → D_1
  26. Translate(C_1, u) → F
  27. Vector(C_1, F) → c_1
  28. Translate(D_1, u) → G
  29. Vector(D_1, G) → d_1
  30. Translate(A, Vector[(-u)]) → A'
  31. Midpoint(C, AlmEnd) → Mid
  32. Segment(C, I) → e
  33. Segment(C', I') → f
  34. Segment(C'', I'') → g
  35. Segment(C''', I''') → h
  36. Point(e, ((n - abs(k)) * 2 / abs(k))) → H
  37. Point(f, ((n - abs(k)) * 2 / abs(k))) → J
  38. Point(g, ((n - abs(k)) * 2 / abs(k))) → K
  39. Point(h, ((n - abs(k)) * 2 / abs(k))) → L
  40. Segment(C'', K) → i
  41. Segment(C''', L) → j
  42. Segment(C, H) → v
  43. Segment(C', J) → w
  44. Segment(I, C') → l
  45. Segment(I', C'') → m
  46. Segment(I'', C''') → p
  47. Segment(I''', C'''') → q
  48. Point(l, ((n - abs(k)) * 2 / abs(k))) → M
  49. Point(m, ((n - abs(k)) * 2 / abs(k))) → N
  50. Point(p, ((n - abs(k)) * 2 / abs(k))) → O
  51. Point(q, ((n - abs(k)) * 2 / abs(k))) → P
  52. Segment(I, M) → r
  53. Segment(I', N) → s
  54. Segment(I'', O) → t
  55. Segment(I''', P) → e_1
  56. Segment(C, Q) → f_1
  57. Segment(Q, EndC) → g_1
  58. Point(f_1, n_1 - 1) → R
  59. Point(g_1, n_1 - 1) → S
  60. Segment(C, R) → h_1
  61. Segment(Q, S) → i_1