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向量间夹角计算

向量间夹角计算
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2016-12-11 09:41
角向量三角函数解析几何几何作图
①本范例演示了如何计算两个向量之间的夹角；②涉及的核心数学概念包括向量、三角函数、向量的模长和点积；③作图要点——从下方『构造摘要』提炼用到了哪些 GeoGebra 命令/对象：CornerThreeD, Polygon, Segment, Sequence, Join, Remove, Point, Vector, Angle, Distance；④适合的学段或用途：高中及以上数学教学，向量与三角函数应用。

GeoGebra 命令及参数

  1. CornerThreeD(-1, 1) → L
  2. CornerThreeD(-1, 7) → U
  3. Polygon((x(L), y(L), 0), (x(L), y(U), 0), (x(U), y(U), 0), (x(U), y(L), 0)) → poly1, f, g, h, i
  4. Segment(O, (0, y(U), 0)) → e
  5. Sequence(Sequence[(p, q, 0), p, ceil(x(L)), floor(x(U)), 1], q, ceil(y(L)), floor(y(U)), 1) → list1
  6. Join(list1) → list2
  7. Remove(list2, {(0, 0, 0)}) → list3
  8. Point(list3) → A
  9. Point(list3) → B
  10. Sequence((x(A), y(A), k), k, ceil(z(L)), floor(z(U)), 1) → list4
  11. Sequence((x(B), y(B), k), k, ceil(z(L)), floor(z(U)), 1) → list5
  12. Point(list4) → C
  13. Point(list5) → D
  14. Vector(O, C) → u
  15. Vector(O, D) → v
  16. Vector((0, 0, 1)) → a
  17. Segment(C, A) → j
  18. Segment(D, B) → k
  19. Angle(D, O, C) → α
  20. Segment(O, A) → l
  21. Segment(O, B) → m
  22. Angle(O, A, C) → β
  23. Angle(O, B, D) → γ
  24. Segment(O, (0, 0, z(U))) → n
  25. Segment(O, (x(U), 0, 0)) → w
  26. Sequence(Segment[(ceil(x(L)), m, 0), (floor(x(U)), m, 0)], m, ceil(y(L)), floor(y(U)), 1) → list7
  27. Sequence(Segment[(n, ceil(y(L)), 0), (n, floor(y(U)), 0)], n, ceil(x(L)), floor(x(U)), 1) → list6
  28. Distance(O, C) → oc
  29. Distance(O, D) → od