Two adjacent sides of a parallelogram ABCD are given by AB =2 i +10 j +11 k and AD =- i +2 j +2 k . The side AD is rotated by an acute angle in the plane of parallelogram so that AD becomes AD ^ . If AD ^ makes a right angle with the side AB then =

Step-by-step solution

Given vectors for adjacent sides of parallelogram ABCD: a = AB = 2 i + 10 j + 11 k b = AD = - i + 2 j + 2 k Magnitudes are | a | = 2^2 + 10^2 + 11^2 = 15 and | b | = (-1)^2 + 2^2 + 2^2 = 3 . After rotation, b' remains in the parallelogram's plane and is orthogonal to a , so we express it as b' = x a + y b with the constraints: 729 a b' = 0 The dot product a b = 40 yields 225x + 40y = 0 y = - 45 8 x . Using | b' |^2 = | b |^2 = 9 : 225x^2 + 9y^2 + 80xy = 9 Substituting y and simplifying gives x^2 = 64 425 x = 8 5 17