Let `D ,Ea n dF` be the middle points of the sides `B C ,C Aa n dA B ,` respectively of a triangle `A B Cdot` Then prove that ` vec A D+ vec B E+ vec C F= vec0` .

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 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

If A B Ca n dA ' B ' C are two triangles and G , G ' be their centriods, prove that vec AA '+ vec B B '+ vec CC '=3 vec G G '

If A, B, C, D be any four points and E and F be the middle points of AC and BD respectively, then A vec(B) + C vec(B) +C vec (D) + vec(AD) is equal to

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

The position vectors of the vertices A ,Ba n dC of a triangle are three unit vectors vec a , vec b ,a n d vec c , respectively. A vector vec d is such that vecd dot vec a= vecd dot vec b= vec d dot vec ca n d vec d=lambda( vec b+ vec c)dot Then triangle A B C is a. acute angled b. obtuse angled c. right angled d. none of these

If A( vec a),B( vec b)a n dC( vec c) are three non-collinear points and origin does not lie in the plane of the points A ,Ba n dC , then point P( vec p) in the plane of the A B C such that vector vec O P is _|_ to planeof A B C , show that vec O P=([ vec a vec b vec c]( vec axx vec b+ vec bxx vec c+ vec cxx vec a))/(4^2),w h e r e is the area of the A B Cdot

If vec a,vec b,vec c represent the sides of a triangle taken in order,then write the value of vec a+vec b+vec *

If vec a , vec b , vec c , vec d are the position vector of point A , B , C and D , respectively referred to the same origin O such that no three of these point are collinear and vec a + vec c = vec b + vec d , than prove that quadrilateral A B C D is a parallelogram.

Prove that the points A,B and C with position vectors vec a,vec b and vec c respectively are collinear if and only if vec a xxvec b+vec b xxvec c+vec c xxvec a=vec 0