Complex parametrisation of a line

Question

Suppose that i have the following planar Cartesian parametrisation of a line $$x=a+tv ~~~\mbox{and }~ y=b+tw$$ where $t$ is the variable parameter and $a$, $b$, $v$ and $w$ are all constants. My question is do i have then $$(a+ib)+t(v+iw)$$ the aformentioned two Cartesian and complex hypothesis are they equivalent?

Answer 1

The two parametrization are ''equivalent'' in the sense that, in the Argand plane, the equation $$1) \qquad z=(a+ib)+t(v+iw)$$ represents the same line as

$2) \qquad z=(x+iy)$ with $x=a+tv$ and $y=b+tw$.

But note that, rigorously, the points $(x,y)$ in $\mathbb{R}^2$ are not the same thing as the complex numbers $z$.