Determining if two planes are parallel with variables?

I've been given two planes. P1 = 2x + ay + 4z = 8 and P2 = ax + 4y +8z = 18

First I was asked to find the normal vectors of both planes:

I found N1 = (2,a,4) and N2 = (a,4,8)

Then I was asked "For what values of 'a' will the two planes be parallel?"

This is where I'm stuck, I cannot find a value of 'a' where the two planes are parallel.

Re: Determining if two planes are parallel with variables?

Quote:

Originally Posted by

**wavingguy** I've been given two planes. P1 = 2x + ay + 4z = 8 and P2 = ax + 4y +8z = 18

First I was asked to find the normal vectors of both planes:

I found N1 = (2,a,4) and N2 = (a,4,8)

Then I was asked "For what values of 'a' will the two planes be parallel?"

This is where I'm stuck, I cannot find a value of 'a' where the two planes are parallel.

Two planes are parallel if their normals are parallel, ie $\displaystyle N_1\|N_2$.

Re: Determining if two planes are parallel with variables?

I understand that, but with the normals I've found there is not value of 'a' that will make the planes parallel. Is it possible that the planes are never parallel?

Re: Determining if two planes are parallel with variables?

Quote:

Originally Posted by

**wavingguy** I understand that, but with the normals I've found there is not value of 'a' that will make the planes parallel. Is it possible that the planes are never parallel?

Well then that is the answer. They cannot be parallel.