What is a sphere ?
If 'C' is a fixed point in space and 'a' is a non-negative real number, then the set of all points 'P' in space, such that the distance CP = a, is called a sphere.
The point 'C' is it's center and 'a' it's radius.
A sphere with a = 0 (i.e, zero radius) is called point sphere.
Another definition : The locus of a variable point in space which moves such that its distance from a fixed point C in space is constant 'a' (≥ 0) is called a sphere.
We shall first discuss Vector and Cartesian equations of a sphere.
The concept of scalar product in vector algebra is used to derive the vector equation.
We shall use bold letters to represent a vector.
Vector form of equation of a sphere
The vector equation of the sphere with center 'C', whose position vector is c, and radius 'a' is given by
| r – c | = a
or r2 – 2r.c + c2 = a2
Conversely, if P(r) is any point satisfying
| r – c | = a, then
CP = a and P lies on the sphere.
If OC = a (i.e, the origin of reference lies on the sphere), the equation of the sphere is
r2 – 2r.c = 0
And if c = 0 (i.e, the origin of reference is the center of the sphere), the equation of the sphere is
r2 = a2 or | r | = a
If A(a) and B(b) are the two end points of a diameter of the sphere, then its vector equation is
(r – a).(r – b) = 0
or r2 – r.(a + b) + a.b = 0