Consider three planes P1, P2 and P3 where a, b and c are constants.
Plane 1 : x + 2y − z = 5
Plane 2 : ax + y + z = −2
Plane 3 : bx − 2y + cz = 11
1) Find the constants a, b and c such that the three planes intersect in a single point
2) For which values of d and h are the two planes P1 and P3 parallel?
All I know is there is a single solution when det does not = 0
And the normal vectors to the planes are:
[1, 2, -1]
[a, 1, 1]
[b, -2, c]
The Attempt at a Solution
How do I start this? I really have no clue what to do for part 1. If there's only a single point of intersection, then the determinant can't equal to 0 right? But I'm stuck as to how I can utilize that to help me solve for the constants..
And for part 2, how do you know when two planes are parallel? Is there an equation for it?
Thanks for any help, I've been kinda stumped on this for a while.