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Sarrus' Rule: Geometric Visualization of 3×3 Determinant Calculation

Sarrus' Rule: Geometric Visualization of 3×3 Determinant Calculation
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2024-09-28 23:26
MatrixGeometric ConstructionGraphing
This example demonstrates Sarrus' rule for calculating a 3×3 determinant through geometric visualization. By mapping matrix elements to directed line segment lengths in a hexagonal diagram, it intuitively shows the calculation process of 'sum of products of main diagonals' minus 'sum of products of anti-diagonals'. Key mathematical concepts include determinant definition, matrix transformation, and geometric interpretation of vector cross product. Construction details: TableText creates the matrix table; Segment draws directed line segments; Circle builds the hexagonal auxiliary diagram; Translate and Rotate handle segment translation and rotation; Length measures segment lengths; Cylinder provides 3D visualization of results. Suitable for high school mathematics (introduction to determinants) and university linear algebra (matrices and determinants), helping students understand the geometric essence of determinant calculation.

GeoGebra Commands & Parameters

  1. TableText(M, "vc", 100, 100) → tab
  2. Segment((-3, 6) + ((-s), s), (3, 0) + (s, (-s))) → d1b
  3. Circle(Point[d1b, t], r) → c1
  4. Segment((-3, 4) + ((-r), r), (3, -2) + (r, (-r))) → d2b
  5. Circle(Point[d2b, t - 2], r) → c2
  6. Translate(c2, (0, 6)) → c2'
  7. Segment((-3, 6), Point[d1b, t]) → r1b
  8. Segment((-3, 2) + ((-r), r), (3, -4) + (r, (-r))) → d3b
  9. Circle(Point[d3b, t - 4], r) → c3
  10. Translate(c3, (0, 6)) → c3'
  11. Segment((-3, 4), If[y(Point[d2b, t - 2]) > 0, Point[d2b, t - 2], (1, 0)]) → r2b
  12. Segment(If[y(Point[d2b, t - 2]) > 0, Point[d2b, t - 2], (1, 6)], (0, 6) + Point[d2b, t - 2]) → r2b'
  13. Segment((-3, 2), If[y(Point[d3b, t - 4]) > 0, Point[d3b, t - 4], (-1, 0)]) → r3b
  14. Segment(If[y(Point[d3b, t - 4]) > 0, Point[d3b, t - 4], (-1, 6)], (0, 6) + Point[d3b, t - 4]) → r3b'
  15. Segment((3, 6) + (s, s), (-3, 0) + ((-s), (-s))) → d4b
  16. Segment((3, 4) + (r, r), (-3, -2) + ((-r), (-r))) → d5b
  17. Segment((3, 2) + (r, r), (-3, -4) + ((-r), (-r))) → d6b
  18. Circle(Point[d4b, t - 6], r) → c4
  19. Segment((3, 6), Point[d4b, t - 6]) → r4b
  20. Circle(Point[d5b, t - 8], r) → c5
  21. Segment((3, 4), If[y(Point[d5b, t - 8]) > 0, Point[d5b, t - 8], (-1, 0)]) → r5b
  22. Segment(If[y(Point[d5b, t - 8]) > 0, Point[d5b, t - 8], (-1, 6)], (0, 6) + Point[d5b, t - 8]) → r5b'
  23. Translate(c5, (0, 6)) → c5'
  24. Circle(Point[d6b, t - 10], r) → c6
  25. Segment((3, 2), If[y(Point[d6b, t - 10]) > 0, Point[d6b, t - 10], (1, 0)]) → r6b
  26. Segment(If[y(Point[d6b, t - 10]) > 0, Point[d6b, t - 10], (1, 6)], (0, 6) + Point[d6b, t - 10]) → r6b'
  27. Translate(c6, (0, 6)) → c6'
  28. Segment((-2.5, 5.5), (2.5, 0.5)) → d1a
  29. Segment((2.5, 5.5), (-2.5, 0.5)) → d4a
  30. Translate(d1a, (0, -2)) → d2a
  31. Translate(d1a, (0, -4)) → d3a
  32. Translate(d4a, (0, -2)) → d5a
  33. Translate(d4a, (0, -4)) → d6a
  34. Segment(Point[d1a, 0], Point[d1a, t]) → r1a
  35. Segment(Point[d2a, 0], Point[d2a, t - 2]) → r2a
  36. Segment(Point[d3a, 0], Point[d3a, t - 4]) → r3a
  37. Segment(Point[d4a, 0], Point[d4a, t - 6]) → r4a
  38. Segment(Point[d5a, 0], Point[d5a, t - 8]) → r5a
  39. Segment(Point[d6a, 0], Point[d6a, t - 10]) → r6a
  40. TableText(M', "vc", 100, 100) → tab'
  41. Translate((-3.06, 6.06), ((0, -6) * Min[t + 1, 1])) → Pu1
  42. Length(t4) → l4
  43. Length(t5) → l5
  44. Length(t6) → l6
  45. CurveCartesian((3 * k / pi) - 2, (-2 * sin(k + (2 * pi / 3))), (2 * cos(k + (2 * pi / 3))), k, 0, ((4 * pi / 3) * Min[1, t - 2])) → h2
  46. CurveCartesian((3 * k / pi) - 2, (-2 * sin(k + (4 * pi / 3))), (2 * cos(k + (4 * pi / 3))), k, 0, ((4 * pi / 3) * Min[1, t - 4])) → h3
  47. CurveCartesian((-3 * k / pi) + 2, (-2 * sin(k)), (2 * cos(k)), k, 0, ((4 * pi / 3) * Min[1, t - 6])) → h4
  48. CurveCartesian((-3 * k / pi) + 2, (-2 * sin(k + (2 * pi / 3))), (2 * cos(k + (2 * pi / 3))), k, 0, ((4 * pi / 3) * Min[1, t - 8])) → h5
  49. CurveCartesian((-3 * k / pi) + 2, (-2 * sin(k + (4 * pi / 3))), (2 * cos(k + (4 * pi / 3))), k, 0, ((4 * pi / 3) * Min[1, t - 10])) → h6
  50. CurveCartesian((3 * k / pi) - 2, (-2 * sin(k)), (2 * cos(k)), k, 0, ((4 * pi / 3) * Min[1, t])) → h1
  51. Rotate((-2, 0, 2), 120°, xAxis) → P_{2,1}
  52. Rotate((0, 0, 2), 120°, xAxis) → P_{2,2}
  53. Rotate((2, 0, 2), 120°, xAxis) → P_{2,3}
  54. Rotate((-2, 0, 2), 240°, xAxis) → P_{3,1}
  55. Rotate((0, 0, 2), 240°, xAxis) → P_{32}
  56. Rotate((2, 0, 2), 240°, xAxis) → P_{3,3}
  57. Cylinder((-2.5, 0, 0), (-2.5 + 5 / 3, 0, 0), 2) → a1, c, d, b
  58. Cylinder((-2.5 + 5 / 3, 0, 0), (-2.5 + 10 / 3, 0, 0), 2) → a2, e, f, g
  59. Cylinder((-2.5 + 10 / 3, 0, 0), (2.5, 0, 0), 2) → a3, h, k, i

AI Drawing Prompt Reference

Create a geometric visualization of Sarrus' rule for calculating a 3×3 determinant. Use directed line segments in a hexagonal diagram to represent matrix element lengths, intuitively showing the calculation process: sum of products along main diagonals minus sum of products along anti-diagonals. Begin with a matrix table, draw multiple directed line segments and auxiliary circles, then translate and rotate them, and finally use curves to provide a 3D visualization of the results.
The prompt above was compiled from the original drawing commands and is for reference only.