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