Linear combination of vector

Question

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

Answer 1

let v=(a,b,c), obviously b= -a-c

is there d and e such that

(d,-d,0)+(0,e,-e) =(d,-d+e,e)=(a,b,c), where b= -a-c

d=a

e=-c

-d+e=-a-c

multiply (1,-1,0) by a and multiply (0,1,-1) by -c

adding equals (a,-a-c,c) which will be on plane

Answer 2

Since v belongs to the plane P we have the relation x+y+z=0 or y=-x-z. So for a=x and b=-z we obtain that the vector v=(x,y,z)=a(1,-1,0)+b(0,1,-1).