Orthogonal Vectors for Sphere Construction: Find Center and Radius

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SUMMARY

The discussion centers on the mathematical representation of a sphere using orthogonal vectors in three-dimensional space. Specifically, the equation (r-a)·(r-b)=0 is established as the condition for orthogonality, which leads to the identification of the sphere's center and radius. The vectors A and B are utilized to demonstrate this geometric principle, paralleling the concept of inscribed angles in a semicircle. The discussion emphasizes the importance of expressing these equations in standard form for clarity.

PREREQUISITES
  • Understanding of vector mathematics and notation
  • Familiarity with the geometric properties of spheres
  • Knowledge of dot product operations in 3D space
  • Basic principles of trigonometry related to angles and circles
NEXT STEPS
  • Study the derivation of the sphere equation from orthogonal vectors
  • Learn about the geometric interpretation of dot products in 3D
  • Explore the implications of inscribed angles in higher dimensions
  • Investigate applications of spheres in computer graphics and physics
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Mathematicians, physics students, and computer graphics developers interested in geometric constructions and vector analysis.

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R <x,y,z>
A<a1,a2,a3>
B<b1,b2,b3>

Show that (r-a).(r-b)=0 represents a sphere find its center and radius

So i see that if 2 vectors are orthogonal you can create a sphere and find the radius and center but can someone better explain this problem
 
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What is there to explain? Write out the equations and put them in standard form to identify it. Geometrically, you may recall that an angle inscribed in a semicircle is a right angle. This is the 3D analogue of that.
 

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