If `A B C D` is a rhombus whose diagonals cut at the origin `O ,` then proved that ` vec O A+ vec O B+ vec O C+ vec O D+ vec Odot`

Let A B C D be a p[arallelogram whose diagonals intersect at P and let O be the origin. Then prove that vec O A+ vec O B+ vec O C+ vec O D=4 vec O Pdot

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.

If 2 vec A C = 3 vec C B , then prove that 2 vec O A =3 vec C B then prove that 2 vec O A + 3 vec O B =5 vec O C where O is the origin.

If vec a , vec b , vec c ,a n d vec d are four non-coplanar unit vector such that vec d make equal angles with all the three vectors vec a , vec ba n d vec c , then prove that [ vec d vec a vec b]=[ vec d vec c vec b]=[ vec d vec c vec a]dot

For any three vectors veca, vec b, vec c prove that (vec a + vec b)+ vec c = vec a + (vec b + vec c)

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 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)

If O( vec0 ) is the circumcentre and O' the orthocentre of a triangle ABC, then prove that i. vec(OA)+vec(OB)+vec(OC)=vec(OO') ii. vec(O'A)+vec(O'B)+vec(O'C)=2vec(O'O) iii. vec(AO')+vec(O'B)+vec(O'C)=2 vec(AO)=vec(AP) where AP is the diameter through A of the circumcircle.

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

A B C D is a quadrilateral. E is the point of intersection of the line joining the midpoints of the opposite sides. If O is any point and vec O A+ vec O B+ vec O C+ vec O D=x vec O E ,t h e nx is equal to a. 3 b. 9 c. 7 d. 4