The vector a + 3 b is perpendicular to 7 a - 5 b and the vector a - 4 b is perpendicular to 7 a - 2 b . Then the angle between a and b is

Step-by-step solution

Given that the vectors are perpendicular, their dot products are zero: ( a + 3 b ) (7 a - 5 b ) = 0 7| a |^2 + 16( a b ) - 15| b |^2 = 0 Also, ( a - 4 b ) (7 a - 2 b ) = 0 7| a |^2 - 30( a b ) + 8| b |^2 = 0 Subtracting the second equation from the first, we get: 46( a b ) - 23| b |^2 = 0 a b = 1 2 | b |^2 Substituting this into the second equation: 7| a |^2 - 30 ( 1 2 | b |^2 ) + 8| b |^2 = 0 7| a |^2 - 15| b |^2 + 8| b |^2 = 0 7| a |^2 - 7| b |^2 = 0 | a | = | b | Let be the angle between a and b . Then, = a b |