Question

find the equation of the plane which passes through P(2,1,-4) and which is perpendicular to both x+y+z=2 and x-y-z=4

Answer 1

The two equations are equations of planes

x + y + z = 2
x - y - z = 4

The normal vectors of these planes are < 1, 1, 1 > and <1 ,-1, -1>.

The normal vector to the plane which contains these two vectors is
< 1, 1, 1 > X < 1 , -1, -1> = < 0, 1, -1 >

(Compute dot products to check this). So the equation of that plane is

y - z = const

P is in the plane so

1 - -4 = 5

We have

0x + y - z = 5