`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` .)

If A ,B ,C ,D be any four points in space, prove that | vec A Bxx vec C D+ vec B Cxx vec A D+ vec C Axx vec B D|=4 (Area of triangle ABC)

A , B , C , D are any four points, prove that vec A Bdot vec C D+ vec B Cdot vec A D+ vec C Adot vec B D=4(Area \ of triangle ABC).

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

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

A,B,C,D are four points in the space and satisfy | vec A B|=3,| vec B C|=7,| vec C D|=11&| vec D A|=9 , then value of vec (AC). vec(BD) .

Let vec a , vec b ,and 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]^2

Prove that [ vec a , vec b , vec c+ vec d]=[ vec a , vec b , vec c]+[ vec a , vec b , vec d]

If veca , vec b , vec c are three vectors such that veca+ vec b+ vec c= vec0 , then prove that vec axx vec b= vec bxx vec c= vec cxx vec a

If vec a , vec b ,a n d vec c are three non-coplanar non-zero vecrtors, then prove that ( vec a . vec a) vec bxx vec c+( vec a . vec b) vec cxx vec a+( vec a . vec c) vec axx vec b=[ vec b vec c vec a] vec a

Let vec a , vec b ,a n d vec c be non-coplanar vectors and let the equation vec a^' , vec b^' , vec c ' are reciprocal system of vector vec a , vec b , vec c , then prove that vec axx vec a^'+ vec bxx vec b^'+ vec cxx vec c ' is a null vector.