Plane goes through points and must be perpendicular to another plane

[RTCriss]

Problem: Write the equation of a plane that goes through the points A(1,1,1) and B(2,2,2) and is perpendicular to the plane: x+y-z=0.
Tried to find vector AB, which gives me(1,1,1), tried to give another random point C(3,4,5) to find normal vector for this plane, got AC(2,3,4). N(normal_vector)=AB x AC = x-2y+z or (1 -2 1). Is correct this method? if it is, how to find the perpendicular plane on x+y-z=0?

 

[Dr.Peterson]

Problem: Write the equation of a plane that goes through the points A(1,1,1) and B(2,2,2) and is perpendicular to the plane: x+y-z=0.
Tried to find vector AB, which gives me(1,1,1), tried to give another random point C(3,4,5) to find normal vector for this plane, got AC(2,3,4). N(normal_vector)=AB x AC = x-2y+z or (1 -2 1). Is correct this method? if it is, how to find the perpendicular plane on x+y-z=0?

If two planes are perpendicular, then their normal vectors are perpendicular.

What justifies your use of a random point C?

There are many planes containing AB (each of whose normal vectors will be perpendicular to AB); you want the one that is perpendicular to x+y-z=0. This gives you two facts from which to find the normal of the desired plane.