Distance between Two Parallel Planes
If ax + by + cz + d1 = 0 and ax + by + cz + d2 = 0 be equation of two parallel planes. Then, the distance between them is
Bisectors of Angles between Two Planes The bisector planes of the angles between the planes a1x + b1y + c1z + d1 = 0, a2x + b2y + c2z + d2 = 0 is a1x + b1y + c1z + d1 / √Σa 2 1 = ± a2x + b2y + c2z + d2 / √Σa 2 2
One of these planes will bisect the acute angle and the other obtuse angle between the given plane.
Sphere
A sphere is the locus of a point which moves in a space in such a way that its distance from a fixed point always remains constant.
General Equation of the Sphere
In Cartesian Form The equation of the sphere with centre( a, b, c) and radius r is
( x – a) 2 +( y – b) 2 +( z – c) 2 = r 2 …….( i) In generally, we can write x 2 + y 2 + z 2 + 2ux + 2vy + 2wz + d = 0 Here, its centre is(-u, v, w) and radius = √u 2 + v 2 + w 2 – d
In Vector Form The vector equation of a sphere of radius a and Centre having position vector c is | r – c | = a