If `A ,B ,C ,D` are four distinct point in space such that `vec(A B)` is not perpendicular to `vec(C D)` and satisfies ` vec (A B).vec (C D)=k(| vec(A D)|^2+| vec(B C)|^2-| vec(A C)|^2-| vec(B D)|^2),` then find the value of `kdot`

Explanation

Let A be the origin, and the position vectors of B,C
and D be `vecb,vecc,vecd`
`vec(AB).vec(CD)=k(|vec(AD)|^(2)+|vec(BC)|^(2)-\|vec(AC)|^(2)-|vec(BD)|^(2))`
or ` (vecb).(vecd - vecc)`
`k = [(vecd)^(2)+(vecc-vecb)^(2)-(vecc)^(2)- (vecd-vecb)^(2)]`
`or vecb.vecd-vecc=k (-2vecb.vecc+ 2vecb .,vecd)`
or k 1/2