The lengths of two opposite edges of a tetrahedron of `aa n db ;` the shortest distane between these edgesis `d ,` and the angel between them if `thetadot` Prove using vector4s that the volume of the tetrahedron is `(a b di s ntheta)/6` .

Explanation

In tetrahedron OABC, take o as the initial point and let the position vectors, of A, B and C be ,`veca, veck and vecc` respectively, then volume of the tetrahedron is equal to `1/6 veca . (veck xx vecc)`
Also `vec(BC) =vecc -veck` so that
`V = 1/6 veca . (veck xx ( veck +vec(BC)) `
` 1/6 veca. (veck xx vec(BC))`
` 1/6 veck. (vec(BC) xxveca)`
` 1/6 veck . |BC||a| sin theta hatn`
Where `hatn` is the unit vector along , PN, the line perpendicular to both OA and BC. Also |BC|=b Here ` V = 1/6 ab sin tehta veck , hatn = 1/6 ab sin theta theta` ( projection of OB on PN)
`1/6 ab sintheta = ` ( perpendicular distance between OA and BC) = `/6 absin theta .d = 1/6 abd sin tehta`