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 parallelogram, and A_1a n dB_1 are the midpoints of sides B Ca n dC D , respectivley . If vec A A_1+ vec A B_1=lambda vec A C ,t h e nlambda is equal to a. 1/2 b. 1 c. 3/2 d. 2 e. 2/3

ABCD is a parallelogram and A_(1) " and " B_(1) are the midpoints of sides BC and CD respectively. If vec(A""A_(1))+vec(AB_(1))=lambda/2vec(AC) , then lambda is equal to -

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 veca = hati+hatj + hat k, vecb=hati+hatj,vec c= hati and (vec axxvecb)xxvec c = lambda vec a+mu vec b . Then show that lambda+mu= 0

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

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 .

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 vec d ⋅ vec a = vec d ⋅ vec b = vec d ⋅ vec c a 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

A , B , Ca n dD are any four points in the space, then prove that | vec A Bxx vec C D+ vec B Cxx vec A D+ vec C Axx vec B D|=4 (area of A B C ).

If vec a , vec b , vec c are non-coplanar vector and lambda is a real number, then the vectors vec a+2 vec b+3 vec c ,lambda vec b+mu vec ca n d(2lambda-1) vec c are coplanar when a. mu in R b. lambda=1/2 c. lambda=0 d. no value of lambda

Let vec a , vec ba n d vec c be three verctors such that vec a!=0,| vec a|=| vec c|=1,| vec b|=4 and | vec bxx vec c|=sqrt(15)dot If vec b-2 vec c=lambda vec a , then find the value of lambda