Let `overset(to)(OA) = overset(to)(a), vec(OB)= 10vec(a) + 2vec(b) " and " overset(to)(OC)=overset(to)(b)` where O,A and C are non-collinear points . Let P denotes the area of the quadrilateral OABC and let q denots the area of the parallelogram with OA and OC as adjacent sides. If p =kq , then k=........

Answer

6

Explanation

Since q= area of parallelogram with `vec(OA) " and " vec(OC)` as adjecent sides `=|vec(OA) xx vec(OC)|=|vec(a) xx vec(b)|`
and p= area of quadrilateral OABC
`= (1)/(2) |vec(OA) xx vec(OB)|+(1)/(2)|vec(OB) xx vec(OC)|`
` = (1)/(2) |vec(a) xx (10vec(a) +2 vec(b) ) + (1)/(2) (10vec(a) + 2vec(b) ) xx vec(b)|`
`= |vec(a) xx vec(b) | + 5 |vec(a) xx vec(b)| = 6|vec(a) xx vec(b)|`
`:. p=6q rArr k=6`