Let a and b be vectors such that | a | = 2 , | a + b | = 19 , and | a - b | = 7 . Then the value of |( a - 2 b ) (2 a + b )|^2 + 25( a b )^2 is equal to ______

Step-by-step solution

First, use the parallelogram law for vectors to find the magnitude of b : | a + b |^2 + | a - b |^2 = 2(| a |^2 + | b |^2) Substitute the given values: ( 19 )^2 + ( 7 )^2 = 2(2^2 + | b |^2) 19 + 7 = 2(4 + | b |^2) 26 = 8 + 2| b |^2 2| b |^2 = 18 | b |^2 = 9 Next, expand the cross product term using distributive properties: ( a - 2 b ) (2 a + b ) = 2( a a ) + ( a b ) - 4( b a ) - 2( b b ) Since a a = 0 , b b = 0 , and b a = -( a b ) , we get: = a b + 4( a b ) = 5( a b ) Squaring the magnitude of this result gives 25