Example: 13  
Find the angle of intersection of the two planes

Solution: 13  
From the equations of the planes, it is evident that the following vectors are to these planes arespectively:
Since the acute angle between the two planes will be the acute angle between their normals, we have
Incidentally, we can now derive the conditions for these planes to be parallel or perpendicular.
Planes are parallel if
Plames are perpendicular if
It should be obvious that for two parallel planes, their equations can be written so that they differ only in the constant term. Thus, any plane parallel to can be written as where

Example: 14  

Solution: 14(a)  
The distance of from the given plane will obviously be measured along the normal to the plane passing through :
We write the equation of the plane as
where is any point on the plane and is the normal to the plane.
Let be the origin. Since lies on the plane, its position vector must satisfy the equation of the plane. But Thus,
Note that where (which sign to take depends on which direction points in).
Thus,

Solution: 14(b)  
Assume any point on the first plane. We have
The distance of from the second plane, say , can be evaluated as described in part above :
Using in we have
This is the required distance between the two planes
