Two adjacent sides of a parallelogram PQRS are given by PQ = j + k and PS = i - j . If the side PS is rotated about the point P by an acute angle in the plane of the parallelogram so that it becomes perpendicular to the side PQ, then ^2 ( 5 2 ) - ^2 ( 2 ) is equal to:

Step-by-step solution

Let be the angle between the vectors PQ and PS . = PQ PS | PQ | | PS | = ( j + k ) ( i - j ) 1^2 + 1^2 1^2 + (-1)^2 = -1 2 2 = - 1 2 = 120^ The side PS is rotated by an acute angle in the plane of the parallelogram to become perpendicular to PQ. The new angle between the sides is 90^ . = 120^ - 90^ = 30^ The given expression is ^2 ( 5 2 ) - ^2 ( 2 ) . Using the trigonometric identity ^2 A - ^2 B = (A+B) (A-B) : ^2 ( 5 2 ) - ^2 ( 2 ) = ( 5 2 + 2 ) ( 5 2 - 2 ) = (3 ) (2 ) Substituting = 30^ : = (90^ ) (60^ ) = 1 3 2