Question

find the vector equation of line passing through the point 2,0,0 and parallel to line r=lambda k

Answer 1

so you want a vector which passes through (2,0,0) and (2,0,λ)
Recall that the line through points with vectors a and b is just
r = a + λ(b-a)
where t is a scalar variable. So in this case,
r = 2i +λk