If `A B C D` is quadrilateral and `Ea n dF` are the mid-points of `A Ca n dB D` respectively, prove that ` vec A B+ vec A D` +` vec C B` +` vec C D` =4 ` vec E Fdot`

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, 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

A,B,C,D are any four points,prove that vec AB*vec CD+BC*vec AD+vec CA*vec BD=0

[vec a, vec b + vec c, vec d] = [vec a, vec b, vec d] + [vec a, vec c, vec d]

In a quadrilateral A B C D , vec A C is the bisector of vec A Ba n d vec A D , angle between vec A Ba n d vec A D is 2pi//3 , 15| vec A C|=3| vec A B|=5| vec A D|dot Then the angle between vec B Aa n d vec C D is cos^(-1)(sqrt(14))/(7sqrt(2)) b. cos^(-1)(sqrt(21))/(7sqrt(3)) c. cos^(-1)2/(sqrt(7)) d. cos^(-1)(2sqrt(7))/(14)

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 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

A straight line L cuts the lines A B ,A Ca n dA D of a parallelogram A B C D at points B_1, C_1a n dD_1, respectively. If ( vec A B)_1,lambda_1 vec A B ,( vec A D)_1=lambda_2 vec A Da n d( vec A C)_1=lambda_3 vec A C , then prove that 1/(lambda_3)=1/(lambda_1)+1/(lambda_2) .

A B C D isa parallelogram with A C a n d B D as diagonals. Then, vec A C- vec B D= 4 vec A B b. 3 vec A B c. 2 vec A B d. vec A B

If vec a,vec b,vec c are the position vectors of points A,B,C and D respectively such that (vec a-vec d)*(vec b-vec c)=(vec b-vec d)*(vec c-vec a)=0 then D is the