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Scalar Multiplication: Vector Scaling and Geometric Visualization

Scalar Multiplication: Vector Scaling and Geometric Visualization
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2016-12-27 11:09
VectorCoordinate GeometryTransformationGeometric ConstructionSlider Animation
In collections Vector Basics
① This example demonstrates the geometric meaning of scalar multiplication: a slider or parameter k controls the scaling of vector u, showing the resulting vector and its position changes in the plane. ② Core mathematical concepts include vector scalar multiplication, collinearity, proportional relationships, and properties of translation and midpoint in plane geometry. ③ Construction highlights: using GeoGebra commands Vector to create vector u, Translate to move points and generate multiple copies (C', C'', C''' etc.), Segment to connect relevant points, Point and Midpoint to determine key locations, and multiple auxiliary objects (like a, b, c, d and their primed versions) to build a complete geometric relationship diagram, with a slider k dynamically showing the effect of scalar multiplication on vector length and direction. ④ Suitable for high school or early college students learning vector algebra and geometric intuition, and also useful for teachers to demonstrate interactive scalar multiplication in class.

GeoGebra Commands & Parameters

  1. Vector(A, B) → u
  2. Translate(C, u) → C'
  3. Translate(C', u) → C''
  4. Translate(C'', u) → C'''
  5. Translate(C''', u) → C''''
  6. Segment(C, C') → a
  7. Segment(C', C'') → b
  8. Segment(C'', C''') → c
  9. Segment(C''', C'''') → d
  10. Segment(A, C) → a'
  11. Segment(A, C') → b'
  12. Segment(A, C'') → c'
  13. Segment(A, C''') → d'
  14. Point(a', n) → A_1
  15. Point(b', n - 1) → B_1
  16. Translate(A_1, u) → D
  17. Vector(A_1, D) → a_1
  18. Translate(B_1, u) → E
  19. Vector(B_1, E) → b_1
  20. Translate(C, Fin) → EndC
  21. Segment(C, EndC) → CEndC
  22. Point(CEndC, ((n - abs(k)) * 2 / abs(k))) → AlmEnd
  23. Vector(C, AlmEnd) → End
  24. Translate(C, Vector[(-u)]) → C'_1
  25. Translate(C'_1, Vector[(-u)]) → C''_1
  26. Translate(C''_1, Vector[(-u)]) → C'''_1
  27. Translate(C'''_1, Vector[(-u)]) → C''''_1
  28. Segment(C, C'_1) → e
  29. Segment(C'_1, C''_1) → f
  30. Segment(C''_1, C'''_1) → g
  31. Segment(C'''_1, C''''_1) → h
  32. Segment(A, C'_1) → e'
  33. Segment(A, C''_1) → f'
  34. Segment(A, C'''_1) → g'
  35. Segment(A, C''''_1) → h'
  36. Point(c', n - 2) → C_1
  37. Point(d', n - 3) → D_1
  38. Translate(C_1, u) → F
  39. Vector(C_1, F) → c_1
  40. Translate(D_1, u) → G
  41. Vector(D_1, G) → d_1
  42. Point(e', n - 1) → E_1
  43. Translate(E_1, Vector[(-u)]) → H
  44. Vector(E_1, H) → e_1
  45. Translate(A_1, Vector[(-u)]) → I
  46. Vector(A_1, I) → a_1'
  47. Point(f', n - 2) → F_1
  48. Point(g', n - 3) → G_1
  49. Translate(F_1, Vector[(-u)]) → J
  50. Vector(F_1, J) → f_1
  51. Translate(G_1, Vector[(-u)]) → K
  52. Vector(G_1, K) → g_1
  53. Translate(A, Vector[(-u)]) → A'
  54. Vector(A, A') → v
  55. Midpoint(C, AlmEnd) → Mid