Let p = a - b , q = a + b . If | a | = | b | = 2 and a b = 2 , then what is the value of | p q | ?

Step-by-step solution

p q = ( a - b ) ( a + b ) = a a + a b - b a - b b Since a a = 0 , b b = 0 , and b a = - a b : p q = 2( a b ) | p q | = 2| a b | Using Lagrange's identity: | a b |^2 = | a |^2 | b |^2 - ( a b )^2 Substituting the given values | a | = 2 , | b | = 2 , and a b = 2 : | a b |^2 = (2)^2 (2)^2 - (2)^2 | a b |^2 = 16 - 4 = 12 | a b | = 12 = 2 3 Therefore, | p q | = 2(2 3 ) = 4 3 Answer: 4 3