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Vector 3D: Geometric Constructions and Spatial Transformations

Vector 3D: Geometric Constructions and Spatial Transformations
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2013-02-03 03:43
VectorCoordinate GeometrySolid GeometryTransformationGeometric Construction
This GeoGebra example demonstrates the geometric construction of 3D vectors and their application in spatial transformations, such as translation and mirroring, to build complex 3D solids like prisms. The core mathematical concepts covered include three-dimensional vectors, spatial coordinate geometry, geometric transformations (translation and reflection), and polygon construction in a 3D coordinate system. To construct these dynamic objects, the example utilizes key GeoGebra commands including Translate, Mirror, Vector, Segment, Line, Sequence, RotateText, Point, Intersect, and Polygon, alongside object lists and interactive buttons to manage coordinates and labels dynamically. It is ideal for high school and university students studying spatial analytic geometry, linear algebra, or vector calculus, serving as an interactive visual tool to bridge abstract vector algebra with concrete 3D geometric shapes.

GeoGebra Commands & Parameters

  1. Translate(O, Vector[u * x_0]) → O_6
  2. Translate(O, Vector[u * y_0]) → O_7
  3. Translate(O, Vector[u * z_0]) → O_8
  4. Mirror(O_6, O) → G
  5. Translate(O, Vector[-0.5 * u * z_0]) → O'_5
  6. Translate(O, Vector[-u * y_0]) → O_2
  7. Vector(O_2, O_7) → v
  8. Vector(G, O_6) → w
  9. Vector(O'_5, O_8) → z_1
  10. Segment(O_2, O_7) → c
  11. Segment(O_6, G) → d
  12. Line(O, O_8) → i
  13. Sequence(O + i * x_0, i, -u, u) → список1
  14. Sequence(O + i * y_0, i, -u, u) → список2
  15. RotateText("\copyright uStas\;\;Strelets", 90*°) → надпись24
  16. Point(w) → M
  17. Point(w) → L
  18. Point(v) → N
  19. Point(v) → P
  20. Line(P, d) → h_1
  21. Line(N, d) → i_1
  22. Line(M, c) → j_1
  23. Line(L, c) → k_1
  24. Intersect(h_1, j_1) → D_2
  25. Intersect(i_1, j_1) → E
  26. Intersect(k_1, i_1) → F
  27. Intersect(h_1, k_1) → H
  28. Polygon(D_2, E, F, H) → многоугольник2, d_2, e_2, f_2, h_2
  29. Vector(A, V) → z
  30. Segment(V, U) → l
  31. Segment(V, U) → m
  32. Segment(B, U) → f
  33. Segment(U, B) → g
  34. Sequence(U + i * z_0, i, 1, z_B) → список1_1
  35. Sequence(U - i * x_0, i, 1, x(B_{xy})) → список5
  36. Sequence(B + i * x_0, i, 1 + x(B_{xy}), -1) → список6
  37. Sequence(U - i * z_0, i, 1, k - 1) → список2_1
  38. Segment(U, R) → e
  39. Segment(R, U) → j
  40. Sequence(U - i * y_0, i, y(B_{xy}), 0) → список7
  41. Segment(A, Q) → n
  42. Segment(Q, I) → p
  43. Segment(Q, D) → q
  44. Segment(Q, A) → r
  45. Segment(Q, I) → s
  46. Segment(D, Q) → t
  47. Sequence(Q + i * z_0, i, 1, z_A) → список3_1
  48. Sequence(Q - i * z_0, i, 1, -z_A - 1) → список8
  49. Sequence(Q - i * x_0, i, 1, x(A_{xy})) → список9
  50. Sequence(I + i * x_0, i, 1 + x(A_{xy}), -1) → список10
  51. Sequence(Q - i * y_0, i, y(A_{xy}), 0) → список11
  52. Sequence(Q - i * y_0, i, 1, y(A_{xy})) → список12
  53. Segment(A_{xy}, C) → d_1
  54. Segment(A_{xy}, B_1) → e_1
  55. Sequence(N + i * x_0, i, -u, u) → список13
  56. Sequence(O_2 + i * x_0, i, -u, u) → список14
  57. Sequence(Segment[Element[список13, i], Element[список14, i]], i, 1, Length[список1]) → список15
  58. Sequence(M + i * y_0, i, -u, u) → список16
  59. Sequence(L + i * y_0, i, -u, u) → список17
  60. Sequence(Segment[Element[список16, i], Element[список17, i]], i, 1, Length[список1]) → список18
  61. Sequence(U - i * y_0, i, 1, y(B_{xy})) → список4
  62. Segment(S, T) → n_1
  63. Segment(T, K) → p_1
  64. Segment(K, J) → q_1
  65. Segment(J, S) → r_1
  66. Segment(S, D_2) → s_1
  67. Segment(K, F) → t_1
  68. Segment(J, H) → a_2
  69. Segment(H, F) → b_2
  70. Segment(T, E) → c_2
  71. Segment(D_2, H) → m_1
  72. Segment(D_2, E) → g_2
  73. Segment(E, F) → i_2
  74. Segment(B_{xy}, W) → h
  75. Segment(B_{xy}, Z) → a_1
  76. Segment(B_{xy}, Z) → j_2
  77. Segment(A_{xy}, C) → k_2
  78. Segment(B_1, A_{xy}) → l_2
  79. Segment(W, B_{xy}) → m_2
  80. Sequence(O + i * z_0, i, -u + 5, u) → список3

AI Drawing Prompt Reference

Please draw a 3D vector geometric construction that demonstrates spatial transformations such as translation and mirroring. First, define the origin O, then perform translations using vectors to obtain multiple points, for example along the x, y, and z axes. Next, apply mirror transformations to these points and construct corresponding vectors. Use these vectors and points to create segments and lines, ultimately forming a complex 3D polygon. The figure also includes auxiliary elements like labels and sequences to help understand spatial relationships.
The prompt above was compiled from the original drawing commands and is for reference only.