Complex parametrization of a line

Question

Suppose that i have a line which passes through the point $A(a+ib)$ and its director vector is $\vec{W}(v+iw)$ then do i have as a complex parametrization of the line the following $$(a+ib)+ t(v+iw)$$ where $t$ is the variable parameter and all the rest $a$ $b$ $v$ and $w$ are constants.

Answer 1

Yes, $$ L(t)= P+tV = (a+ib)+ t(v+iw)$$ is a parametrization of a straight line passing through $P=(a+bi)$ at $t=0$ with the direction vector of $V=(v+iw)$

Answer 2

Although you have complex numbers, the way you use vectors is still the same as in the x,y plane. So yes, this should be correct.