Prove that in a tetrahedron if two pairs of opposite edges are perpendicular , then the third pair is also perpendicular.

Explanation

Let ABCD be the tetratahedron and A be at the origion.
`vec(AB)=vecb,vec(AC)=veccandvec(AD)=vecd`
Let the edge AB be perendicualar to the oppostie edge BD. Therefore,
`vec(AB).vec(CD)=0`
`vecb.(vecd-vecc)=0`
` or vecb.vecd=vecb.vecc`
Also let AC be perpendicular to the opposite egde BD. therefore,
`vec(AC).vec(BD)=0`
`vecc.(vecd-vecb)=0`
`vecc.vecd=vecb.vecc`
Now from (i) and (ii) we have `vecb.vecd=vecc.vecd`
`(vecc-vecb).vecd=0`
`or vec(BC). vec(AD)=0`
Hence, AD is perpendicular to opposite edge BC.