Volume of tetrahedron formed by the planes `x+y=0, y+z=0, z+x=0,x+y+z-1=0` is

Options

A. `(1)/(6)`
B. `(1)/(3)`
C. `(2)/(3)`
D. None of these

Answer

C

Explanation

Solving the planes taking 3 at a time
Vertices of tetrahedron are
`O: (0,0,0), A: (1,1,-1),B:(1,-1,1),C:(-1,1,1)`
`implies "Volume"=(1)/(3) xx("area of base")xx("height")`
`=(1)/(3)xxsqrt(3)/(4)xx(2sqrt(2))^(2)xx(1)/sqrt(3)=(2)/(3)`
Or Volume `=(1)/(6)[vec(a) vec(b) vec(c)]`
`("where" vec(a)=vec(OA), vec(b)=vec(OB) and vec(c) = vec(OC))`