The vector equation of the line passing through the points (4,2,3) and which is parallel to the vector i +2j -5k is given by-

(4+λ)i + 2(1+λ)j + (3-5λ) k 

We have vector a = 4i+ 2j +3k and vector b= i+2j -5k

Therefore the vector equation of the line is

vector r   = a + λb

         ⇒ r = (4i+ 2j +3k ) + λ (i+2j -5k)

         ⇒ r = (4+λ)i + 2(1+λ)j + (3-5λ) k