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`

If | vec a|+| vec b|=| vec c|a n d vec a+ vec b= vec c , then find the angle between vec aa n d vec bdot

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 .)

If vec a , vec b ,a n d vec c are mutually perpendicular vectors and vec a=alpha( vec axx vec b)+beta( vec bxx vec c)+gamma( vec cxx vec a)a n d[ vec a vec b vec c]=1, then find the value of alpha+beta+gammadot

Given | vec a|=| vec b|=1a n d| vec a+ vec b|=sqrt(3). If vec c is a vector such that vec c- vec a-2 vec b=3( vec axx vec b), then find the value of vec c dot vec b

If three unit vectors vec a , vec b ,a n d vec c satisfy vec a+ vec b+ vec c=0, then find the angle between vec aa n d vec bdot

If vec a , vec ba n d vec c are unit vectors satisfying | vec a- vec b|^2+| vec b- vec c|^2+| vec c- vec a|^2=9, then |2 vec a+5 vec b+5 vec c| is.

If vectors vec aa n d vec b are two adjacent sides of a parallelogram, then the vector respresenting the altitude of the parallelogram which is the perpendicular to a is vec b+( vec bxx vec a)/(| vec a|^2) b. ( vec a . vec b)/(| vec b|^2) c. vec b-( vec b . vec a)/(| vec a|^2) d. ( vec axx( vec bxx vec a))/(| vec b|^2)

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

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 .