Evaluating surface integral (solving for unknown variables)

[Miike012]

The solutions have came up with 5 equations, I'm not confused how they got those 5 equations but I don't understand how it was concluded that L = 0 and m = p = 1/√2.

 

[LCKurtz]

The solutions have came up with 5 equations, I'm not confused how they got those 5 equations but I don't understand how it was concluded that L = 0 and m = p = 1/√2.

They massaged the equations somehow, who knows? The obvious way to work that problem is to get a normal vector by crossing the vectors along two adjacent sides. (Unless you see how to write the equation of the plane by inspection).

 

[Miike012]

Nevermind, I figured it out.. But if someone has a better way please let me know.
This is my solution.
Lx + my + pz = 2p
Lx + my + pz = 2L + 2p
Lx + my + pz = 2L + 2m
Lx + my + pz = 2m
L^2 + m^2 + p^2 = 1

Then I looked at the right side of the first four equations and noticed they are all satisfied only if L = 0...
However I know this solution probably won't work for more difficult problems...
So please someone post a better solution

 

[Miike012]

They massaged the equations somehow, who knows? The obvious way to work that problem is to get a normal vector by crossing the vectors along two adjacent sides. (Unless you see how to write the equation of the plane by inspection).

Dang why didn't I think of that.. that's the easiest solution... just cross two vectors.. Thank you.