Vector in plane question

Question

let P be the plane x+y+z=0. show that any linear combination of v₁=(1,-1,0) and v2=(0,1,-1) is in P

Answer 1

Consider a linear combination of the two given vectors, that is a(1,-1,0)+b(0,1,-1)=(a,b-a,-b). For x=a, y=b-a and z=-b we have that x+y+z=a+b-a-b=0 as desired.