`A B C D` parallelogram, and `A_1a n dB_1` are the midpoints of sides `B Ca n dC D ,` respectivley . If ` vec AA_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`

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 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

G is the centroid of triangle A B Ca n dA_1a n dB_1 are rthe midpoints of sides A Ba n dA C , respectively. If "Delta"_1 is the area of quadrilateral G A_1A B_1a n d"Delta" is the area of triangle A B C , then "Delta"//"Delta"_1 is equal to a. 3/2 b. 3 c. 1/3 d. none of these

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 .

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) .

If vec aa n d vec b are two unit vectors incline at angle pi//3 , then { vec axx( vec b+ vec axx vec b)}dot vec b is equal to a. (-3)/4 b. 1/4 c. 3/4 d. 1/2

Let vec a , vec b a n d vec c be three vectors having magnitudes 1, 5and 3, respectively, such that the angel between vec aa n d vec bi stheta and vec axx( vec axx vec b)=c . Then t a ntheta is equal to a. 0 b. 2/3 c. 3/5 d. 3/4

ABCDE is a pentagon .prove that the resultant of force vec A B , vec A E , vec B C , vec D C , vec E D and vec A C ,is 3 vec A C .

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

vec a , vec b ,a n d vec c are three vectors of equal magnitude. The angel between each pair of vectors is pi//3 such that | vec a+ vec b+ vec c|=6. Then | vec a| is equal to 2 b. -1 c. 1 d. sqrt(6)//3