u , v , w are three unit vectors. Let p = u + v + w , q = u ( v w ) .If p u = 3 2 , p v = 7 4 ,| p |=2 and v = K q , then K=

Step-by-step solution

Given that | u |=| v |=| w |=1 p u =( u + v + w ) u = 3 2 | u |^2+ u v + w u = 3 2 u v + w u = 3 2 -1= 1 2 ....(i) p v =( u + v + w ) v = 7 4 u v +| v |^2+ v w = 7 4 u v + v w = 7 4 -1= 3 4 ....(ii) | p |^2=| u + v + w |^2 4=| u |^2+| v |^2+| w |^2+2( u v + v w + w u ) 1 2 = u v + v w + w u ....(iii) On solving (i), (ii) and (iii), we get u v = 3 4 , v w =0 and w u =- 1 4 v =k q =k[ u ( v w )]=k[( u w ) v -( u v ) w ] v =k (- 1 4 v - 3 4 w ) 4 v =-k v -3 k w (4+k) v =-3 k w |4+k|| v |=|-3 k|| w | |4+k|=|3 k| 4+k=3