Linear Algebra - Help with Planes

Homework Statement

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?

Homework Equations

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.

Er, anybody? Sorry to bump.

If the planes intersect at a single point then there is a single solution to the augmented matrix, form the augmented matrix and row reduce, you should see the light upon figuring that out.

Edit - if the planes are parallel then that means there is no solution to the system. OR, if they are parallel then they have the same normal vector (or a multiple of it)

If the planes intersect at a single point then there is a single solution to the augmented matrix, form the augmented matrix and row reduce, you should see the light upon figuring that out.

Edit - if the planes are parallel then that means there is no solution to the system. OR, if they are parallel then they have the same normal vector (or a multiple of it)

Hey thanks for taking the time to answer. The thing is, I JUST got into this course, and everything is still very new to me. The professor hasn't even ever mentioned what an augmented matrix is or what row reduce means. Do you think you can elaborate a bit? Thanks..

Char. Limit
Gold Member
Hey thanks for taking the time to answer. The thing is, I JUST got into this course, and everything is still very new to me. The professor hasn't even ever mentioned what an augmented matrix is or what row reduce means. Do you think you can elaborate a bit? Thanks..

That's odd, because in the linear algebra course I'm taking right now, those concepts are introduced in the first week. Your augmented matrix looks like this:

$$$\left( \begin{array}{cccc} 1 & 2 & -1 & 5 \\ a & 1 & 1 & -2 \\ b & -2 & c & 11 \end{array} \right)$$$

You might have learned row-reduction under the name Gaussian elimination, perhaps.