[Math] Determine a scalar equation is that perpendicular to the line of intersection of the planes $2x + y – z + 5 = 0$ and $x + y + 2z + 7 = 0.$

Question

Determine a scalar equation for the plane that passes through the point $(2, 0, −1)$ and is perpendicular to the line of intersection of the planes
$$2x + y – z + 5 = 0 \;\text{ and }\; x + y + 2z + 7 = 0.$$

Answer 1

it's simple. the normal vector of the plane is found from cross product of 2 given planes normal vectors: $\vec n=\vec n_1 \times \vec n2 = (2,1,-1) \times (1,1,2) = (3,-5,1)$

then with the point given you can write the equation of the plane perpendicular to those planes like this:

assume $(x,y,z)$ is on desired plan and the point $(2,0,-1)$ is also on the plane. so the vector formed by this two point must be perpendicular to normal of plane that we got with cross product so theie dot product is equal to zero: $$(x-2,y,z+1).(3,-5,1)=0 \Rightarrow 3x-5y+z=5$$