a , b are non-collinear vectors, | a |=2 2 ,| b |=3 and the angle between a and b is 45^ . Then, the lengths of the diagonals of the parallelogram whose adjacent sides are represented by the vectors 5 a +2 b and a -3 b are

Step-by-step solution

aligned p & =5 a +2 b q & = a -3 b B D & = p + q =6 a - b C A & = q - p =-4 a -5 b aligned array r Now, | B D |=|6 a - b | 36 a +| b |^2-12 a b array aligned & = 36| a |^2+| b |^2-12| a || b | 45^ & = 36 8+9-12 2 2 3 1 2 & = 225 =15 aligned and | C A |=|-(4 a +5 b )|= |4 a |^2+|5 b |^2+40| a || b | 45^ aligned & = 16 8+25 9+40 2 2 3 1 2 & = 593 aligned Lengths of the diagonals are 15 and 593 .