Let a = ( - 1)t i + j + (3 - 2) k and b = t i + 2t j + k be two vectors, where t R and is a real parameter. If the vector projection of a on b is a non-zero vector in the direction of b for all t R , then the smallest integer value of is

Step-by-step solution

The vector projection of a on b is given by ( a b | b |^2 ) b . For this to be a non-zero vector in the direction of b , we must have a b > 0 for all t R . Computing the dot product: a b = ( - 1)t(t) + (2t) + (3 - 2)(1) > 0 ( - 1)t^2 + 2 t + (3 - 2) > 0 For this quadratic in t to be strictly positive for all t R , two conditions must be satisfied: 1. Leading coefficient must be positive: - 1 > 0 > 1 2. Discriminant must be negative: D (2 )^2 - 4( - 1)(3 - 2) 4 ^2 - 4(3 ^2 - 5 + 2) -2 ^2 + 5 - 2 2 ^2 - 5 + 2 > 0 (2