( iv ) Four points A ( x1 , y1 , z1 ), B ( x2 , y2 , z2 ), C ( x3 , y3 , z3 ) and D ( x4 , y4 , z4 ) are coplanar if and only if
( v ) Equation of the plane containing two coplanar lines
Angle between Two Planes
The angle between two planes is defined as the angle between the normal to them from any point .
Thus , the angle between the two planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0
is equal to the angle between the normals with direction cosines ± a1 / √Σ a 2 1 , ± b1 / √Σ a 2 1 , ± c1 / √Σ a 2 1
and ± a2 / √Σ a 2 2 , ± b2 / √Σ a 2 2 , ± c2 / √Σ a 2 2
If θ is the angle between the normals , then cos θ = ± a1a2 + b1b2 + c1c2 / √a 2 1 + b 2 1 + c 2 1 √a 2 2 + b 2 2 + c 2 2