Let a and b be two vectors forming the adjacent sides of a parallelogram. The angle between a and b is acute, | a | = 2 , and a b = 3 | a b | . If c = 3 ( a b ) - b and | c | = 8 , then the area of the parallelogram formed by a and b is

Step-by-step solution

Given a b = 3 | a b | | a || b | = 3 | a || b | = 1 3 = 6 The area of the parallelogram is | a b | . | a b | = | a || b | ( 6 ) = 2| b | ( 1 2 ) = | b | Now, c = 3 ( a b ) - b Since a b is perpendicular to b , we have ( a b ) b = 0 . | c |^2 = 3| a b |^2 + | b |^2 Substitute | a b | = | b | : | c |^2 = 3| b |^2 + | b |^2 = 4| b |^2 Given | c | = 8 , we get 64 = 4| b |^2 | b |^2 = 16 | b | = 4 . Thus, the area of the parallelogram is | a b | = | b | = 4 . Answer: 4