Hi, here's a question from an exam that I don't know how to do. I can do (i) and (ii). For (ii) I get $\displaystyle z_2-z_1$ or $\displaystyle z_1-z_2$. But I don't know what they want for (iii) and (iv).

I was thinking of using the fact that the diagonals of a paralleogram bisect each other, but that doesn't seem to work out...

(i) Let $\displaystyle z_1=r_1 cis \theta_1$ and $\displaystyle z_2=r_2 cis \theta_2$. If $\displaystyle z_1$ and $\displaystyle z_2$ are parallel then prove that $\displaystyle z_1 = k z_2$ for k real.

(ii) Let the points A, B, C and D be represented by the complex numbers $\displaystyle z_1$, $\displaystyle z_2$, $\displaystyle z_3$ and $\displaystyle z_4$ respectively (in clockwise order). Find two possible vectors representing side AB.

(iii) If A, B, C and D are the vertices of a parallelogram then using parts (i) and (ii) find an expression for side DC in terms of AB using the vectors $\displaystyle z_1$, $\displaystyle z_2$, $\displaystyle z_3$ and $\displaystyle z_4$.

(iv) Using a property of a parallelogram find the two possible values of k.

(v) Show that for both these values of k that $\displaystyle z_1-z_2-z_3+z_4=0$.

Any ideas?

Thanks

EDIT: Does this belong in the pre calculus section? If so, many apologies. My university considers complex numbers as an algebra topic...