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If A ,B ,C ,D are four distinct point in...

If `A ,B ,C ,D` are four distinct point in space such that `A B` is not perpendicular to `C D` and satisfies ` vec A Bdot 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`

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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
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