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3D Vectors and Plane Geometry Practice (H2M L11 Q15)

3D Vectors and Plane Geometry Practice (H2M L11 Q15)
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2020-02-12 19:36
AngleVectorCoordinate GeometrySolid GeometryGeometric Construction
This example demonstrates a dynamic geometry problem combining 3D vectors, planes, rotations, and random number generation to explore spatial line-plane relationships and vector operations. Core mathematical concepts include 3D vectors (e.g., OA, AB, BC), plane equations, orthogonal planes, point-to-plane distances, angles (α, β), greatest common divisor (GCD), and random integer sequences. Construction highlights: using Random, Sequence, Remove, Sample, Element, Plane, Angle, If, GCD, Distance, Length, OrthogonalPlane, Line, Solutions, and TableText commands; objects include buttons (button1, button2), sliders (nr, nd, etc.), points (A, B, C, D, O), lines (l1, l2, l3), planes (p), and text labels. Suitable for high school or early college solid geometry and vector instruction, this can serve as classroom exploration or homework, helping students understand vector decomposition, plane relationships, and dynamic construction in 3D space.

GeoGebra Commands & Parameters

  1. Random(5, 5) → nr
  2. Sequence(Sequence[Sequence[If[(i * n1) + (j * n2) + (k * n3) ≟ nd, (i, j, k), (0, 0, 0)], i, (-nr), nr], j, (-nr), nr], k, (-nr), nr) → l1
  3. Remove(Unique[Flatten[l1]], {(0, 0, 0)}) → l2
  4. Sample(l2, 3, false) → l3
  5. Element(l3, 2) → B
  6. Element(l3, 1) → A
  7. Element(l3, 3) → C
  8. Plane(C, B, D) → q
  9. Angle(p, q) → α
  10. If(α > 90°, 180° - α, α) → β
  11. GCD({x(vBC), y(vBC), z(vBC)}) → fbc
  12. GCD({x(vAB), y(vAB), z(vAB)}) → fab
  13. Distance(D, p) → ddp
  14. Length(l2) → nl2
  15. GCD({x(vAC), y(vAC), z(vAC)}) → fac
  16. OrthogonalPlane(A, Vector[vAB ⊗ vAC]) → p1
  17. Line(B, vBC1) → f
  18. GCD({x(vDB), y(vDB), z(vDB)}) → fdb
  19. Line(D, vDB1) → g
  20. Solutions((z(vBC1) * x) = (-z(B))) → l4
  21. TableText(l5, "c( )") → text6
  22. If(z(vBC1) ≟ 0, text7, text5) → text8

AI Drawing Prompt Reference

Draw a 3D vector and plane geometry figure. First, randomly generate three points A, B, and C, then construct a plane q through these points. Calculate the angle α between planes p and q, and take its supplementary angle β. Compute the greatest common divisors fbc, fab, and fac of the coordinates of vectors AB, AC, and BC. Determine the distance ddp from point D to plane p. Construct a plane p1 perpendicular to plane p through point A and vector vAB×vAC. Construct a line f through point B and vector vBC, and a line g through point D and vector vDB. Finally, find the intersection of lines f and g and display the result.
The prompt above was compiled from the original drawing commands and is for reference only.