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Lines & Planes: Vector Forms in 3D Space

Lines & Planes: Vector Forms in 3D Space
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2019-01-17 03:24
VectorCoordinate GeometrySolid GeometryMatrixParametric Equation
① This example demonstrates how to represent lines and planes in 3D space using vector forms, with interactive visualization of their geometric relationships. ② Core mathematical concepts include parametric vector equations of lines, point-normal form of planes, vector operations (dot and cross products), and algebraic representation of spatial geometric elements. ③ Construction highlights: GeoGebra commands such as Vector, Plane, PointIn, Line, Sequence, and Point are used; objects include multiple origins (Origin_B, Origin_F, Origin_H), points A, E, T1, plane objects BrownPlaneOrigin_B and BrownPlane_A, a matrix M_matrix, and a button for interaction. ④ Suitable for high school advanced or undergraduate students learning analytic geometry and linear algebra, and for teachers as a classroom demonstration tool to visualize the correspondence between vector forms and spatial figures.

GeoGebra Commands & Parameters

  1. Vector(Origin, G) → T1
  2. Plane(A, Origin_B, Origin) → a
  3. PointIn(a) → E
  4. PointIn(a) → Origin_F
  5. Vector(Origin_B, A) → T1_A
  6. Line(qwInitPt, wTermPt) → Line_{2D}
  7. Vector(qwInitPt, wTermPt) → w
  8. Vector(qwInitPt, qwInitPt + (t * w)) → wScl
  9. Vector(O, qwInitPt) → u
  10. Vector(O, qwInitPt + (t * w)) → v
  11. Plane(O, qwInitPt, wTermPt) → n
  12. Vector(qwInitPt, qTermPt) → q
  13. Vector(qwInitPt, qwInitPt + (s * q)) → qScl
  14. Line(qwInitPt, qTermPt) → f_1
  15. Vector(O, qTermPtScl) → v_1
  16. Plane(wTermPt, qwInitPt, qTermPt) → c_1
  17. Sequence(Line[qwInitPt + (t * w), q], t, -15, 15) → qParallelLinesPlane
  18. Sequence(qwInitPt + (t * w), t, -15, 15) → wCoordPts
  19. Sequence(qwInitPt + (s * q), s, -15, 15) → qCoordPts
  20. Sequence(Line[qwInitPt + (s * q), w], s, -15, 15) → wParallelLinesPlane
  21. Sequence(Text[" " + (k), qwInitPt + (k * q)], k, -15, 15) → qAxisNumberList
  22. Vector(wTermPt, wTermPt + Vector[(1, 0, -3)]) → u_1
  23. Vector(wTermPt, wTermPt + Vector[(1, 2, -3)]) → w_1
  24. Vector(wTermPt, wTermPt + Vector[(1, -1, -3)]) → h_1
  25. Point(qwInitPt, Vector[(s * q) + (t * w)]) → GridPt
  26. Sequence(Text[" " + (k), qwInitPt + (k * w)], k, -15, 15) → wAxisNumberList