Complex Area of parallelogram

Question

$(O,\vec u, \vec v)$ is a direct orthonormal system. $I(1+i); K(2+2i); M(2+2e^{i\alpha});N(2i-2e^{i\alpha})\\$ 1)Show that $OMKN$ is a parallelogram and that $$Area(OMKN)= 4[1+\sqrt2cos(\alpha+\frac\pi4)]$$

Answer 1

$e^{ia} = \cos \theta + i\sin \theta$

Lets represent these in more traditional cartesian coordinates.

$M = (2 + 2\cos a, 2\sin a)\\ N = (- 4\cos a, 2-2\sin a)\\ K = M+N$

That is a parallelogram.

As for the Area. The "shoelaces algorithm" tells us.

$A = (2 + 2\cos a, 2\sin a) \times (- 4\cos a, 2-2\sin a)$

$(2+2\cos a)(2-2\sin a) + 4\sin\cos a = 4 +4\cos a - 4\sin a = 4(1+\cos a + \sin a)$

and

$\cos a - \sin a = \sqrt 2 (\cos \frac {\pi}{4} \cos a - \sin \frac {\pi}{4}\sin a) = \sqrt 2\cos (a+ \frac {\pi}{4})$