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Vector Algebra: Intersections of Line and Half-Circle

Vector Algebra: Intersections of Line and Half-Circle
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2013-02-21 13:53
VectorCoordinate GeometryLocus
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This example demonstrates how to determine the intersection point between a ray parallel to an initial ray and a half-circle using GeoGebra. Key mathematical concepts include vectors, coordinate geometry, geometric construction, and locus. It utilizes commands and objects such as Intersect, Circle, Point, Vector, Ray, OrthogonalLine, Line, Angle, Segment, CircleArc, Locus, and If. Suitable for high school and above students exploring vector interactions with geometric shapes.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → O
  2. Circle(O, 1) → unitCircle
  3. Point(unitCircle) → N
  4. Vector(O, N) → n
  5. Point(unitCircle) → L
  6. Vector(O, L) → l
  7. Ray(Lamp, O) → p1
  8. OrthogonalLine(L, p1) → p
  9. Point(p) → F
  10. Vector(L, F) → u
  11. Vector(O, -δ * Perp * l) → d
  12. Vector(-Perp * n) → e
  13. Line(O, N) → north
  14. Angle(n, l) → θ
  15. Angle(C, O, D) → ξ
  16. Vector(G, A) → z
  17. Angle(O, G, A) → κ
  18. Segment(W, E) → f
  19. CircleArc(O, W, E) → block
  20. Line(O, E) → east
  21. Locus(H, t) → loc1
  22. Locus(dFB, F) → loc2
  23. Locus(dBC, F) → loc4
  24. Locus(dFC, F) → loc5
  25. Locus(dAC, F) → loc6
  26. Locus(dFA, F) → loc7
  27. If(abs(δ) ≤ 1, If[NA < 0, A, If[abs(δ) < abs(N * L), C, Q]], Q) → J
  28. Segment(F, J) → s
  29. Segment(Lamp, If[NL > 0, O, L]) → h
  30. Segment(M, P) → k
  31. Vector(O, L) → v
  32. Segment(a + 0.25 * Perp * a, a - 0.25 * Perp * a) → R
  33. Vector(A, S) → w
  34. Angle(S, A, F) → θ'
  35. If(NA < 0, true, false) → Abelow
  36. If(NL < 0, true, false) → Lbelow

AI Drawing Prompt Reference

Draw the intersection point between a ray parallel to an initial ray and a half-circle, demonstrating vectors, coordinate geometry, geometric construction, and locus. Start by plotting the axes intersection, draw a unit circle centered at the origin, take two points on the circle to create vectors, then draw a parallel ray intersecting the circle. Find the intersection point and create vectors from the origin and between points. Next, draw a perpendicular line, find the foot of the perpendicular, and create vectors. Finally, plot loci and connecting segments based on conditions.
The prompt above was compiled from the original drawing commands and is for reference only.