# Scaling and rotating complex numbers as vectors

## Scaling complex numbers

When a complex number $$z$$ is multiplied by a positive real number, then it scales the length of the vector, keeping the same direction. If it is multiplied by a negative real number, then it reverses the direction and also scales the length.

Example. Let $$z=1+i$$. Then we can plot $$2z$$, $$-3z$$ and $$0.5z$$ as vectors.

## Rotating complex numbers

When a complex number $$z$$ is multiplied by $$r(\cos\theta+i\sin\theta)$$, it is rotation of $$\theta$$ counter-clockwise and the length is multiplied by $$r$$.

Similarly, when a complex number $$z$$ is divided by $$r(\cos\theta+i\sin\theta)$$, it is rotation of $$\theta$$ clockwise and the length is divided by $$r$$.

Example. When a vector is multiplied by $$i$$, it is a 90 degree rotation counterclockwise with no change in modulus.

Worked exercise. Let $$z=re^{i\theta}$$, where $$0<\theta<\frac{\pi}{2}$$. Represent $$z$$, $$-iz$$, $$\overline{z}$$ and $$(\sqrt{3}+i)z$$ as vectors on the complex plane.
Solution. $$-iz$$ is a clockwise rotation of $$z$$ by 90 degrees. $$\overline{z}$$ is a reflection of $$z$$ in the real axis. $$\sqrt{3}+i = 2e^{\frac{i\pi}{6}}$$, so $$(\sqrt{3}+i)z$$ is a 30 degree rotation counter-clockwise, followed by a scaling by 2.

## Problems

1. Let $$z=1+2i$$. Plot the following complex numbers as vectors on the complex plane.
1. $$z$$
2. $$-z$$
3. $$\overline{z}$$
4. $$3z$$
5. $$iz$$
2. $$z_1$$, $$z_2$$ and $$z_3$$ are three complex numbers such that $$z_2-z_1=i(z_3-z_1).$$ If points $$P$$, $$Q$$ and $$R$$ represent the complex numbers $$z_1$$, $$z_2$$ and $$z_3$$ respectively, what geometric properties does the triangle $$PQR$$ have?
3. Let $$O$$ be the point that represents the complex number $$0$$, and points $$P$$ and $$Q$$ represent the complex numbers $$z_1$$ and $$z_2$$ respectively. Triangle $$OPQ$$ is equilateral. Show that $$z_1^2+z_2^2=z_1z_2$$.
4. Hard. Let $$z$$ and $$w$$ be complex numbers such that $$\frac{z-w}{z+w}$$ is imaginary. Show geometrically that $$|z|=|w|$$.