Two vectors are given as a = 2 i + 2 j + k and b = x i + 4 j - 4 k , where x > 0 . The vector b is rotated by an acute angle in the plane containing a and b such that it becomes perpendicular to a . If = 5 3 , then the value of x is:

Step-by-step solution

Let be the initial angle between the vectors a and b . Since b is rotated by an acute angle to become perpendicular to a , the new angle between them is 90^ . Therefore, the initial angle must satisfy | - 90^ | = , which implies = 90^ . Taking the cosine on both sides: = (90^ ) = Given = 5 3 and is acute, we find = 1 - ( 5 3 )^2 = 2 3 . Thus, = 2 3 . Using the dot product formula for : = a b | a | | b | a b = (2)(x) + (2)(4) + (1)(-4) = 2x + 4 | a | = 2^2 + 2^2 + 1^2 = 3 | b | = x^2 + 4^2 + (-4)^2 = x^2 + 32 Since