If a parallelogram is constructed on the vectors a = 3 p - q , b = p + 3 q and | p | = 3, | q | = 2 and angle between p and q is 3 , then the ratio of the lengths of adjacent sides a and b of the parallelogram is

Step-by-step solution

Given a = 3 p - q and b = p + 3 q . We are given | p | = 3 , | q | = 2 , and the angle between p and q is 3 . The dot product of p and q is: p q = | p | | q | ( 3 ) = 3 2 1 2 = 3 The length of side a is given by: | a |^2 = |3 p - q |^2 = 9| p |^2 + | q |^2 - 6( p q ) | a |^2 = 9(3)^2 + (2)^2 - 6(3) = 81 + 4 - 18 = 67 | a | = 67 The length of side b is given by: | b |^2 = | p + 3 q |^2 = | p |^2 + 9| q |^2 + 6( p q ) | b |^2 = (3)^2 + 9(2)^2 + 6(3) = 9 + 36 + 18 = 63 | b | = 63 The ratio of the lengths of adjacent s