Let the vectors a = 3 i + 4 k and b represent two adjacent sides of a parallelogram. If the area of the parallelogram is 10 3 and the projection of b along a is 2 , then the sum of the squares of the lengths of the diagonals of the parallelogram is equal to

Step-by-step solution

Given a = 3 i + 4 k , its magnitude is: | a | = 3^2 + 0^2 + 4^2 = 5 The area of the parallelogram is given by the magnitude of the cross product of its adjacent sides: | a b | = 10 3 The projection of b along a is given as 2 : a b | a | = 2 a b 5 = 2 a b = 10 Using Lagrange's identity relating the dot product and cross product: | a |^2 | b |^2 = | a b |^2 + ( a b )^2 Substituting the known values: (5)^2 | b |^2 = (10 3 )^2 + (10)^2 25 | b |^2 = 300 + 100 = 400 | b |^2 = 16 The lengths of the diagonals of the parall