`A , B , Ca n dD` are any four points in the space, then prove that `| vec A Bxx vec C D+ vec B Cxx vec A D+ vec C Axx vec B D|=4` (area of ` A B C` .)

Explanation

Let the position vectors of points A,B,C,D be `veca, vecb, vecc and vecd`, respectively, with respect to some origin.
`|vec(AB)xxvec(CD)+vec(BC)xx vec(AD)+vec(CA)xx vec(BD)|`
`[|(vecb-veca)xx(vecd-vecc)+ (vecc-vecb) xx (vecd -veca) + (veca -vecc) xx (vecd-vecb)|`
`2|vecb xx veca +vecc xx vecb +vecaxxvecc|`
`2(2 xx ("area of " triangleABC)`
` 4xx (" area of " triangle ABC)`