`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 , 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 ).

Prove that [[ vec a+ vec b, vec b+ vec c, vec c+ vec a]]=2[ [vec a, vec b, vec c]]dot

ABCDE is a pentagon. Prove that the resultant of force vec A B , vec A E , vec B C , vec D C , vec E D and vec A C ,is 3 vec A C .

Given the vectors vec A , vec B ,a n d vec C form a triangle such that vec A= vec B+ vec Cdot find a ,b ,c ,a n dd such that the area of the triangle is 5sqrt6 where vec A=a hat i+b hat j+c hat k vec B=d hat i+3 hat j+4 hat k vec C=3 hat i+ hat j-2 hat k

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

If vec a , vec ba n d vec c are non-coplanar vectors, prove that the four points 2 vec a+3 vec b- vec c , vec a-2 vec b+3 vec c ,3 vec a+ 4 vec b-2 vec ca n d vec a-6 vec b+6 vec c are coplanar.

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

Let vec a , vec ba n d vec c be unit vectors, such that vec a+ vec b+ vec c= vec x , vec a dot vec x=1, vec b dot vec x=3/2,| vec x|=2. Then find the angle between vec c and vec x

The condition for equations vec rxx vec a= vec ba n d vec rxx vec c= vec d to be consistent is a. vec bdot vec c= vec adot vec d b. vec adot vec b= vec c dot vec d c. vec bdot vec c+ vec adot vec d=0 d. vec adot vec b+ vec c dot vec d=0

For any four vectors, vec a , vec b , vec c and vec d prove that vec d.( vec axx( vec bxx( vec cxx vec d)))=( vec b. vec d)[ vec a \ vec c \ vec d] .