## Cases of Intersecting Planes

Whether or not three planes intersect in points, lines or coincide depends on the relationships between the equations of the planes The equations of the planes are multiples of each other. For example   The second and third equations are double and triple the first respectively. The equations of two planes are multiples of each other and the third is some other. For example   The second is twice the first. Two of the planes are multiples of each other and but the third is not because the number on the right hand side fails.    Twow of the planes are parallel so the coeeficients are the same or multiples of each other and the third is not.    All planes are parallel so the coefficients are multiples of each other but not the right hand side.    All the equations are completely independent of each other.    The equations are linearly dependent, so each equation can be written in terms of each other.   The first equation minus the second equation equations the third equation Each pair of planes intesects and the line of intersection is parallel to the third plane. There is no common point of intersection.   The second equation minus 3 times the first gives and the third minus the first gives The equations are inconsistent. 