IS this right?

find the parametrization of the line joining $\displaystyle z_0 = 1 $ and $\displaystyle z_1 = 2 + i $

pick x(t) = t

then $\displaystyle 1 <= t <= 2 $

and

y(t) is a linear function of t so y(t) = at + b

when t = 1, y = 0 ; so 0 = a + b

when t = 2, y = 1; so 1 = 2a + b

after subtracting,

-1 = -a

so a = 1, then b = -1

so $\displaystyle z(t) = t + i(t - 1) $

thank you