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Perpendicularity of a Line and a Plane

Perpendicularity of a Line and a Plane
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2016-05-16 11:31
VectorSolid GeometryGeometric Construction
In collections Projections and Angles
This example demonstrates the geometric relationship and determination theorem of perpendicularity between a line and a plane. Core mathematical concepts include the definition of line-plane perpendicularity, vector method for judgment, the three-perpendiculars theorem, and point-line distance formulas in analytical geometry. Construction highlights involve GeoGebra commands/objects like Intersect (intersection), Point (point), CircumcircleArc (arc), Line (line), Segment (segment), Circle (circle), Angle (angle), and Translate (translation). Suitable for high school solid geometry teaching and geometric construction practice, helping students intuitively understand abstract concepts.

GeoGebra Commands & Parameters

  1. Intersect(yAxis, zAxis) → A
  2. Point(zAxis) → C
  3. Point(xAxis) → B
  4. CircumcircleArc(B, C, (-4, 0, 0)) → c
  5. Point(c) → P
  6. Line(A, P) → i
  7. Intersect(yAxis, zAxis) → D
  8. Segment(A, P) → f
  9. Circle(i, (0, -2, 0)) → d
  10. Point(d) → E
  11. Segment(A, E) → g
  12. Angle(E, A, P) → α
  13. Angle(B, A, P) → β
  14. Segment(A, A + (2, 0, 0)) → h
  15. Translate(Polygon[(7, 7), (-7, 7), (-7, -7), (7, -7)], Vector[(0, 0)]) → poly1
  16. Segment(A, (0, -2, 0)) → j
  17. Angle((0, -2, 0), A, P) → γ

AI Drawing Prompt Reference

Draw the perpendicular relationship between a line and a plane in a 3D coordinate system. Start by setting up the coordinate axes, then take point A on the z-axis and point B on the x-axis. Construct an arc c with B and C as endpoints, and take point P on the arc. Draw line i through A and P, intersecting the y-axis at point D. Construct segments AP, AE, and AE', and calculate the angles ∠EAP, ∠BAE', and ∠EAE'. Finally, translate the quadrilateral and construct segment AE', and calculate the angle ∠E'AE.
The prompt above was compiled from the original drawing commands and is for reference only.