If A,B, C and D are four distinct points in space that `vec(AB)` is not perpendicular to `vec(CD)` and satisfies `(AB).(CD)=k(|vec(AD)|^(2)+ |vec(BC)|^(2)- |vec(BD)|^(2))`, then find the value of k .

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