ABCD is a quadrilateral with AB = a , AD = b and AC =2 a +3 ~b . If its area is times the area of the parallelogram with AB , AD as adjacent sides, then the value of is

Step-by-step solution

Define AB = a and AD = b . The area of the parallelogram formed by these vectors is A_ parallelogram = | a b | . The quadrilateral ABCD is divided by diagonal AC = 2 a + 3 b into two triangles: ABC and ADC . The area of ABC is 1 2 | a (2 a + 3 b )| = 1 2 |3( a b )| = 3 2 | a b | , as a a = 0 . The area of ADC is 1 2 | b (2 a + 3 b )| = 1 2 |2( b a )| = 1 2 | -2( a b ) | = | a b | , using b b = 0 and b a = -( a b ) . Summing these gives the total area: 3 2 | a b | + | a b | = 5 2 | a b | . Given that this equals | a