The area of a parallelogram whose diagonals are the vectors 2 a - b and 4 a -5 b , where a and b are unit vectors forming an angle of 45^ is

Step-by-step solution

The diagonals of the parallelogram are given by d ₁ = 2 a - b and d ₂ = 4 a - 5 b , where a and b are unit vectors separated by 45^ . The area of a parallelogram with diagonals d ₁ and d ₂ is 1 2 | d ₁ d ₂| . The cross product evaluates to: d ₁ d ₂ = (2 a - b ) (4 a - 5 b ) = -10( a b ) + 4( a b ) = -6( a b ) Taking magnitude gives | d ₁ d ₂| = 6| a b | . Since | a b | = | a || b | (45^ ) = 1 1 1 2 = 1 2 , it follows that | d ₁ d ₂| = 6 2 = 3 2 . The area is therefore 1 2 3 2 = 3 2 2 . This matches option B .