Let O (0, 0) , A (-1, 2) and B (1, 3) be the vertices of OAB. The bisector of angle O intersects side AB at point D. The value of OD AB is equal to...

Step-by-step solution

Position vectors of the vertices are OA = - i + 2 j and OB = i + 3 j . The lengths of the sides OA and OB are: | OA | = (-1)^2 + 2^2 = 5 | OB | = 1^2 + 3^2 = 10 By the angle bisector theorem, the point D divides the segment AB in the ratio of the adjacent sides: AD DB = | OA | | OB | = 5 10 = 1 2 Using the section formula, the position vector of D is: OD = 2 OA + 1 OB 2 + 1 = 2 (- i + 2 j ) + ( i + 3 j ) 2 + 1 = (1 - 2 ) i + (2 2 + 3) j 2 + 1 The vector AB is given by: AB = OB - OA = ( i + 3 j ) - (- i + 2 j ) = 2