If a and b are two vectors such that | a | = 2 and | b | = 3 , then the maximum value of 3 | (3 a + 2 b ) | + 4 | (3 a - 2 b ) | is :

Step-by-step solution

Let be the angle between a and b . We have |3 a + 2 b | = 9| a |^2 + 4| b |^2 + 12| a || b | Substituting | a | = 2 and | b | = 3 : |3 a + 2 b | = 9(4) + 4(9) + 12(2)(3) = 36 + 36 + 72 |3 a + 2 b | = 72(1+ ) = 144 ^2 ( 2 ) = 12 ( 2 ) Similarly, |3 a - 2 b | = 9| a |^2 + 4| b |^2 - 12| a || b | |3 a - 2 b | = 36 + 36 - 72 = 72(1- ) = 144 ^2 ( 2 ) = 12 ( 2 ) The given expression becomes: 3|3 a + 2 b | + 4|3 a - 2 b | = 3 (12 ( 2 ) ) + 4 (12 ( 2 ) ) = 36 ( 2 ) + 48 ( 2 ) The maximum value of A x + B x is A^2 + B^2 . M