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45° Rotation Matrix in Surd Form

45° Rotation Matrix in Surd Form
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2016-05-10 18:31
AngleTrigonometryMatrixTransformation
① This example illustrates the matrix for a 45-degree rotation, expressed in surd form. ② It involves key concepts such as matrix representation, rotation transformations, trigonometric values (sin45°=√2/2, cos45°=√2/2), and radical calculations. ③ The construction uses GeoGebra commands including Circle, Point, Angle, Line, OrthogonalLine, Intersect, Polygon, Midpoint, Segment, and If, building a geometric model and displaying the matrix via text objects. ④ Designed for high school or first‑year university students studying linear algebra and analytic geometry, and suitable for instructional demonstrations of deriving rotation matrices.

GeoGebra Commands & Parameters

  1. Circle(A, C) → c
  2. Point(c) → B
  3. Angle(C, A, B) → α
  4. Line(B, A) → f
  5. OrthogonalLine(A, f) → g
  6. Intersect(c, g, 1) → D
  7. Polygon(D, A, B) → poly1, b, d, a
  8. Midpoint(D, B) → E
  9. Segment(A, E) → h
  10. Midpoint(A, E) → F
  11. Midpoint(F, E) → G
  12. Intersect(p, n) → H
  13. Intersect(m, p) → I
  14. Intersect(m, q) → J
  15. Intersect(n, q) → K
  16. If(y(B) < 0, "-", "") → a_3
  17. If(x(B) < 0, "-", " ") → a_1
  18. If(y(D) < 0, "-", "") → a_4
  19. If(t, If[85° < α < 95°, false, If[175° < α < 185°, false, If[265° < α < 275°, false, If[α < 5°, false, If[355° < α, false, If[Mod[α, 45°] < °, true, If[Mod[α, 45°] > 44°, true, false]]]]]]], false) → e
  20. Midpoint(B, E) → L
  21. Midpoint(D, E) → N
  22. Segment(E, B) → r
  23. Segment(D, E) → s
  24. If(x(D) < 0, "-", "") → a_2
  25. If(e, false, If[t, true, false]) → o

AI Drawing Prompt Reference

Draw a diagram illustrating a 45-degree rotation matrix, featuring a circle and points on it, showing matrix representation through rotation and geometric constructions, with labeled angles and segments, and text objects displaying matrix elements, reflecting sine/cosine values and radical calculations, suitable for linear algebra instruction.
The prompt above was compiled from the original drawing commands and is for reference only.