Question

Find the volume of the tetrahedron formed by planes whose equations are y+z=0, z+x=0, x+y=0,and x+y+z=1

Answer 1

Find the vertices, which are the planes intersections, i.e..

x+y=0; y+z=0; z+x=0→A(0,0,0)

x+y=0; y+z=0; x+y+z−1=0→B(1,−1,1)

x+y=0; z+x=0; x+y+z−1=0→C(−1,1,1)

y+z=0; z+x=0; x+y+z−1=0→D(1,1,−1)

AB=(1,−1,1); AC=(−1,1,1); AD=(1,1,−1)

Now the volume is given by 

V=1/6 [AB AC AD]

Thus, volume is 2/3