`A , B , C , D` are any four points, prove that ` vec A B dot vec C D+ vec B Cdot vec A D+ vec C Adot vec B D=0.`

A , B , Ca n dD are any four points in the space, then prove that | vec A Bxx vec C D+ vec B Cxx vec A D+ vec C Axx vec B D|=4 (area of A B C .)

A B C D E is pentagon, prove that vec A B + vec B C + vec C D + vec D E+ vec E A = vec0 vec A B+ vec A E+ vec B C+ vec D C+ vec E D+ vec A C=3 vec A C

For any four vectors, prove that ( vec bxx vec c)dot( vec axx vec d)+( vec cxx vec a)dot( vec bxx vec d)+( vec axx vec b)dot( vec cxx vec d)=0.

If vec aa n d vec b are two vectors, then prove that ( vec axx vec b)^2=| vec adot vec a vec adot vec b vec bdot vec a vec bdot vec b| .

If vectors vec a , vec b ,a n d vec c are coplanar, show that | vec a vec b vec c vec adot vec a vec adot vec b vec adot vec c vec bdot vec a vec bdot vec b vec bdot vec c|=0

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

Let vec a , vec b ,a n d vec c be any three vectors, then prove that [ vec axx vec b vec bxx vec c vec cxx vec a]=[ vec a vec b vec c]^2dot

If vec a , vec b ,a n d vec c be three non-coplanar vector and a^(prime),b^(prime)a n dc ' constitute the reciprocal system of vectors, then prove that vec r=( vec rdot vec a ') vec a+( vec rdot vec b^') vec b+( vec rdot vec c^') vec c vec r=( vec rdot vec a ') vec a '+( vec rdot vec b^') vec b '+( vec rdot vec c ') vec c '

If A B C D is quadrilateral and Ea n dF are the mid-points of A Ca n dB D respectively, prove that vec A B+ vec A D + vec C B + vec C D =4 vec E Fdot

If vec a , vec ba n d vec c are three non coplanar vectors, then prove that vec d=( vec adot vec d)/([ vec a vec b vec c])( vec bxx vec c)+( vec bdot vec d)/([ vec a vec b vec c])( vec cxx vec a)+( vec cdot vec d)/([ vec a vec b vec c])( vec axx vec b)