If `D` is the mid-point of the side `B C` of a triangle `A B C ,` prove that ` vec A B+ vec A C=2 vec A Ddot`

If D is the mid-point of the side BC of a triangle ABC, prove that vec AB+vec AC=2vec AD .

If O is a point in space, A B C is a triangle and D , E , F are the mid-points of the sides B C ,C A and A B respectively of the triangle, prove that vec O A + vec O B+ vec O C= vec O D+ vec O E+ vec O Fdot

If a,b,c are the lengths of sides,BC,CA and AB of a triangle ABC, prove that vec BC+vec CA+vec AB=vec O and deduce that (a)/(sin A)=(b)/(sin B)=(c)/(sin C) .

If vec a,vec b,vec c and vec d are the position vectors of the vertices of a cyclic quadrilateral ABCD prove that (|vec a xxvec b+vec b xxvec d+vec d xxvec a|)/((vec b-vec a)*(vec d-vec a))+(|vec b xxvec c+vec c xxvec d+vec d xxvec b|)/((vec b-vec c)*(vec d-vec c))=

If vec a,vec b,vec c are the position vectors of the vertices A,B,C of a triangle ABC, show that the area triangle ABCis(1)/(2)|vec a xxvec b+vec b xxvec c+vec c xxvec a| Deduce the condition for points vec a,vec b,vec c to be collinear.

If vec a,vec b,vec c be the vectors represented by theside sof a triangle,taken in order,then prove that vec a+vec b+vec c=vec 0

In a triangle OAC, if B is the mid point of side AC and vec O A= vec a , vec O B= vec b , then what is vec O C ?

If vec a,vec b and vec c are the position vectors of the vertices A,B and C respect ively,of bigoplus ABC , prove that the perpendicular distance of the vertedx A from the base BC of the triangle ABC is (|vec a xxvec b+vec b xxvec c+vec c xxvec a|)/(|vec c-vec b|)

If A(vec a);B(vec b) and C(vec c) are three non collinear points,then for any point P(vec p) in the plane of the Delta ABC, prove that [vec a,vec b,vec c]=vec p,(vec axvec b+vec bxvec c+vec cxvec a)

If the vectors vec a , vec b ,a n d vec c form the sides B C ,C Aa n dA B , respectively, of triangle A B C ,t h e n vec adot vec b+ vec bdot vec c+ vec cdot vec a=0 b. vec axx vec b= vec bxx vec c= vec cxx vec a c. vec adot vec b= vec bdot vec c= vec cdot vec a d. vec axx vec b+ vec bxx vec c+ vec cxx vec a=0