Proving Orthogonality of Planes and Lines in 3D Geometry

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SUMMARY

This discussion centers on proving the orthogonality of planes and lines in 3D geometry, specifically involving planes A, B, and C. Plane A is formed by points v, w, and x, while plane B is formed by points w, x, and y. Plane C is orthogonal to both A and B and passes through points v and y. The participants conclude that the line formed by the intersection of plane C with plane B is indeed orthogonal to line wx, supported by vector geometry and the properties of normal vectors.

PREREQUISITES
  • Understanding of 3D geometry and spatial reasoning
  • Familiarity with vector operations, particularly the cross product
  • Knowledge of the properties of orthogonal planes and lines
  • Basic skills in constructing geometric proofs
NEXT STEPS
  • Study the properties of normal vectors in 3D geometry
  • Learn about the cross product and its applications in determining orthogonality
  • Explore geometric proofs involving multiple planes and their intersections
  • Investigate the implications of orthogonality in higher-dimensional spaces
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Mathematicians, geometry enthusiasts, students studying 3D geometry, and anyone interested in understanding the relationships between planes and lines in three-dimensional space.

  • #31
gnome said:
They're all just lines.

Oh, 'X' means cross product. I thought you meant something else by FXG, etc., the lack of spacing threw me.

Then what you wrote is axiomatic, and it follows from the fact that we're working in 3 dimensions. Simple enough ? :smile:
 
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  • #32
I guess. It just bothers me that I can't put my finger on any specific definition or postulate that says that.
 
  • #33
Draw a perpendicular from point w to the plane C. Any plane that contains that line is, by definition, perpendicular to C.
In particular, the plane that contains that perpendicular and the intersection of A and C. Since a point and a line define a plane, this plane must be plane A.
Now, consider the plane that contains the perpendicular and the intersection of B and C. By the same reasoning, this must be plane B.
So, the perpendicular from w to C belongs simultaneously to A and B and must be their intersection.
The intersection of A and B is orthogonal to all lines in C. QED.
 

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