state parallelogram law of vector addition. derive analytically expression for magnitude and direction of resultant vector 

when 2 vectors are represented as adjescent side of a parallelogram , the diagonal indicate the resultant  vector

let boac be the parallelogram , ob -vector b and oa - vector a

extend a tod and join c to d

boa =theta and coa = alpha

oc be the diagonal  ie , resultant vector -r

then, in ocd

oc2 = od2 +cd2

ie, r2 = (oa+ad)2 +cd2

  = oa2+2oa *ad +ad2 + cd2  (a+b)2

in acd, ad2+cd2= ac2  - ( 1)

r2= oa2+2oa.ad+ac2

in acd, cos theta = ad / ac

 ad=ac cos theta 

vectorb cos theta

ac= vector b ,  ab = ac

  (1) =  vec a2 + 2abcostheta+b2