Vector form of equation of a sphere
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A point P(r), other than A and B, lies on the sphere.
AP.BP = 0 (∵ ∠APB = 90°)
(r – a).(r – b) = 0
r2 – r.(a + b) + a . b = 0
If P = A or B, then AP = 0 or BP = 0.
Hence the equation of the sphere with A(a) and B(b) as the end points of a diameter is
(r – a).(r – b) = 0
or r2 – r.(a + b) + a.b = 0