A parallelogram is constructed on 5 a + 2 b and a - 3 b as its adjacent sides, with | a | = 2 2 , | b | = 3 . The angle between a and b is 4 . Then the length of the diagonals of the parallelogram are

Step-by-step solution

Let the adjacent sides of the parallelogram be u = 5 a + 2 b and v = a - 3 b . The diagonals of the parallelogram are given by the sum and difference of the adjacent sides: d ₁ = u + v = (5 a + 2 b ) + ( a - 3 b ) = 6 a - b d ₂ = u - v = (5 a + 2 b ) - ( a - 3 b ) = 4 a + 5 b Given | a | = 2 2 , | b | = 3 , and the angle between a and b is 4 . The dot product of a and b is: a b = | a | | b | ( 4 ) = (2 2 )(3) ( 1 2 ) = 6 Now, finding the length of the first diagonal: | d ₁|^2 = |6 a - b |^2 = 36| a |^2 - 12( a b )