If | a |=2 2 and | b |=3 and angle between a and b is 4 . If a parallelogram is constructed with adjacent sides p =2 a -3 b and q = a + b then the product of length of both the diagonals is

Step-by-step solution

Given | a | = 2 2 , | b | = 3 , and the angle = 4 between a and b . The dot product is a b = | a | | b | ( 4 ) = (2 2 )(3) ( 1 2 ) = 6 . The diagonals of the parallelogram are d₁ = p + q = (2 a - 3 b ) + ( a + b ) = 3 a - 2 b and d₂ = p - q = (2 a - 3 b ) - ( a + b ) = a - 4 b . Calculate | d₁ |^2 = |3 a - 2 b |^2 = 9| a |^2 + 4| b |^2 - 12( a b ) = 9(8) + 4(9) - 12(6) = 72 + 36 - 72 = 36 . Thus, | d₁ | = 6 . Calculate | d₂ |^2 = | a - 4 b |^2 = | a |^2 + 16| b |^2 - 8( a b ) = 8 + 16(9) - 8(6) = 8 + 144 - 48 = 104