If the vectors a = c ( ₇ x ) i +2 j +3 k and b = ( ₇ x ) i +3 c ( ₇ x ) j -4 k make obtuse angle for any x>0 , then c belongs to

Step-by-step solution

Let a = c( ₇ x) i + 2 j + 3 k and b = ( ₇ x) i + 3c( ₇ x) j - 4 k . For an obtuse angle between vectors, their dot product must be negative: a b Substitute y = ₇ x R to obtain a = cy i + 2 j + 3 k and b = y i + 3cy j - 4 k . The dot product becomes: a b = (cy)(y) + (2)(3cy) + (3)(-4) = cy^2 + 6cy - 12 We require cy^2 + 6cy - 12 c (6c)^2 - 4(c)(-12) Factoring yields 12c(3c + 4)