Let a , b , and c be three non-zero vectors such that | a | = 2 , | b | = 3 , and | c | = 4 . It is given that | a + b + c | = | a + b - c | , | a - c | = | a + c | , and | a b | = 3 ( a b ) . The volume of the parallelepiped whose coterminous edges are represented by the vectors a , b , and c is

Step-by-step solution

From the condition | a - c | = | a + c | , squaring both sides gives: | a |^2 + | c |^2 - 2( a c ) = | a |^2 + | c |^2 + 2( a c ) 4( a c ) = 0 a c = 0 . From the condition | a + b + c | = | a + b - c | , squaring both sides gives: | a + b |^2 + | c |^2 + 2(( a + b ) c ) = | a + b |^2 + | c |^2 - 2(( a + b ) c ) . This implies 4(( a + b ) c ) = 0 a c + b c = 0 . Since a c = 0 , we get b c = 0 . Thus, c is perpendicular to both a and b . From the condition | a b | = 3 ( a b ) , let be the angle between a and b . Then