`A B C D` is a parallelogram. If `La n dM` are the mid-points of `B Ca n dD C` respectively, then express ` vec A La n d vec A M` in terms of ` vec A Ba n d vec A D` . Also, prove that ` vec A L+ vec A M=3/2 vec A Cdot`

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

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 E is pentagon, prove that vec A B + vec B C + vec C D + vec D E+ vec E A = vec0 vec A B+ vec A E+ vec B C+ vec D C+ vec E D+ vec A C=3 vec A C

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 .

Given that vec adot vec b= vec adot vec c , vec axx vec b= vec axx vec ca n d vec a is not a zero vector. Show that vec b= vec cdot

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 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 aa n d vec b are two vectors, then prove that ( vec axx vec b)^2=| vec adot vec a vec adot vec b vec bdot vec a vec bdot vec b| .

If vec a , vec b ,a n d vec c are three vectors such that vec axx vec b= vec c , vec bxx vec c= vec a , vec cxx vec a= vec b , then prove that | vec a|=| vec b|=| vec c|dot

Find | vec a|a n d| vec b|,if( vec a+ vec b)dot( vec a- vec b) = 8 , | vec a|=8| vec b|dot