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Vector Equality in Pyramid Geometric Transformations

Vector Equality in Pyramid Geometric Transformations
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2016-12-26 08:18
PolygonVectorSolid GeometryTransformationInequality
This GeoGebra example dynamically demonstrates vector equality within the context of pyramid (polyhedron) geometric transformations. The core mathematical concepts include vector equality (same magnitude and direction), geometric transformations (translation, rotation), and area continuity. Construction highlights: using Polygon commands to build the base and lateral faces of the pyramid, Vector commands to define vectors u, v and their counterparts, If commands for conditional display, and multiple objects (A, B, C, D and subscripted copies) to create corresponding vector pairs. Sliders or dynamic parameters allow observing how areas change while vectors remain equal. Suitable for high school or early college students learning vector operations, transformations, and area relationships, as well as for teachers demonstrating the intuitive meaning of vector equality.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C, D) → Box1_1, a, b, c, d
  2. Polygon(A_1, B_1, C_1, D_1) → Box2_1, a_1, b_1, c_1, d_1
  3. Polygon(A_2, B_2, C_2, D_2) → Box3_1, a_2, b_2, c_2, d_2
  4. Vector(E, F) → u
  5. Vector(E_1, F_1) → u_1
  6. Vector(E_2, F_2) → u_2
  7. Vector(G, H) → v
  8. Vector(G_1, H_1) → v_1
  9. Vector(G_2, H_2) → v_2
  10. Vector(I, J) → w
  11. Vector(I_1, J_1) → w_1
  12. Vector(I_2, J_2) → w_2
  13. Vector(K, L) → e
  14. Vector(M, N) → f
  15. Vector(O, P) → g
  16. Vector(Q, R) → h
  17. Vector(S, T) → i
  18. Vector(U, V) → j
  19. If(R ∈ Box1_1 ∨ Q ∈ Box1_1 ∨ V ∈ Box1_1 ∨ U ∈ Box1_1 ∨ P ∈ Box1_1 ∨ O ∈ Box1_1 ∨ L ∈ Box1_1 ∨ K ∈ Box1_1 ∨ S ∈ Box1_1 ∨ T ∈ Box1_1 ∨ N ∈ Box1_1 ∨ M ∈ Box1_1, 1, 0) → Box1
  20. If(R ∈ Box2_1 ∨ Q ∈ Box2_1 ∨ V ∈ Box2_1 ∨ U ∈ Box2_1 ∨ P ∈ Box2_1 ∨ O ∈ Box2_1 ∨ L ∈ Box2_1 ∨ K ∈ Box2_1 ∨ S ∈ Box2_1 ∨ T ∈ Box2_1 ∨ N ∈ Box2_1 ∨ M ∈ Box2_1, 1, 0) → Box2
  21. If(R ∈ Box3_1 ∨ Q ∈ Box3_1 ∨ V ∈ Box3_1 ∨ U ∈ Box3_1 ∨ P ∈ Box3_1 ∨ O ∈ Box3_1 ∨ L ∈ Box3_1 ∨ K ∈ Box3_1 ∨ S ∈ Box3_1 ∨ T ∈ Box3_1 ∨ N ∈ Box3_1 ∨ M ∈ Box3_1, 1, 0) → Box3
  22. If(H_1 ∈ Box1_1 ∨ G_1 ∈ Box1_1 ∨ H_2 ∈ Box1_1 ∨ G_2 ∈ Box1_1 ∨ I_1 ∈ Box1_1 ∨ I_2 ∈ Box1_1 ∨ J_1 ∈ Box1_1 ∨ J_2 ∈ Box1_1, 1, 0) → Box'_1
  23. If(E_1 ∈ Box2_1 ∨ E_2 ∈ Box2_1 ∨ F_1 ∈ Box2_1 ∨ F_2 ∈ Box2_1 ∨ I_1 ∈ Box2_1 ∨ I_2 ∈ Box2_1 ∨ J_1 ∈ Box2_1 ∨ J_2 ∈ Box2_1, 1, 0) → Box'_2
  24. If(E_1 ∈ Box3_1 ∨ E_2 ∈ Box3_1 ∨ F_1 ∈ Box3_1 ∨ F_2 ∈ Box3_1 ∨ H_1 ∈ Box3_1 ∨ H_2 ∈ Box3_1 ∨ G_1 ∈ Box3_1 ∨ G_2 ∈ Box3_1, 1, 0) → Box'_3

AI Drawing Prompt Reference

Draw a 3D pyramid scene with three base polygons of different sizes stacked at varying heights. Each polygon has corresponding vertices, and multiple vector pairs are drawn between points, with matching vectors across layers having identical direction and length to show vector equality. Additionally, include auxiliary vectors inside or on the pyramid surface, and use conditional logic to highlight when certain points fall within a specific base polygon. The composition emphasizes vector invariance under spatial transformations, suitable for educational demonstration.
The prompt above was compiled from the original drawing commands and is for reference only.