Math Help - complex number

1. complex number

A and B are the points representing the complex numbers z and (1 + i )z in an Argand diagram. Prove that triangle OAB is a right-angled isoceles triangle, where O is the origin.

2. Re: complex number

Hints: $|OA|=|z|,\;|OB|=\sqrt{2}|z|$ . On the other hand $|AB|=|(1+i)z-z|=|iz|=|z|$ and $1+i=\sqrt{2}e^{\pi i/4}$ .

3. Re: complex number

Given:
$OA = z, OB = (1 + i)z$
By the definition of multiplication of complex numbers if $z_1 = r_1. cis {\theta}_1, z_1 = r_2. cis {\theta}_2$ we have
$z_1.z_2 = r_1.r_2 cis({\theta}_1 + {\theta}_2)$ so we have $z_1 = (1 + i) = \sqrt 2 .cis {\frac{\pi}{4}}$.
$\therefore$ in $\Delta OAB, |OB| = \sqrt 2.|OA|, \; \angle AOB = \frac{\pi}{4}$

$\therefore \Delta OAB$ is right isosceles triangle.

Kalyan