The problem provided concerns complex numbers, projections, and reflections in the complex plane. Let's break it down step by step and formulate the general solution.

Given:

• $w$ is a complex number with $|w| = 1$, meaning it's a point on the unit circle in the complex plane.
• The line $l_w = \{aw : a \in \mathbb{R} \}$ represents a line passing through the origin in the direction of $w$.
• $z_o$ is the orthogonal projection of a complex number $z$ onto the line $l_w$.
• $z_r$ is the reflection of $z$ with respect to the line $l_w$.

We are asked to express $z_o$ and $z_r$ in terms of $z$ and $w$, using complex arithmetic (addition, multiplication, conjugation).

Step 1: Case where $w = 1$

This represents the real axis, so $l_w$ is simply the real axis. In this case:

• The projection of $z = x + iy$ onto the real axis is $z_o = x$, where $x$ is the real part of $z$. This can be written as: $z_o = \text{Re}(z) = \frac{z + \overline{z}}{2}$
• The reflection of $z$ with respect to the real axis is $z_r = x - iy$. This can be written as: $z_r = \overline{z}$ So for $w = 1$, we have:
• $z_o = \frac{z + \overline{z}}{2}$
• $z_r = \overline{z}$

Step 2: General case for arbitrary $w$

For a general unit complex number $w$, the line $l_w$ is rotated from the real axis by the argument of $w$.

Orthogonal Projection $z_o$:

The orthogonal projection of $z$ onto the line in the direction of $w$ can be given by: $z_o = \frac{z \cdot w^{-1} + \overline{z \cdot w^{-1}}}{2} w$ Here, we rotate the system so that $w$ points along the real axis, apply the real-axis projection, and then rotate back by multiplying by $w$.

This simplifies to: $z_o = \text{Re}(z \cdot \overline{w}) \cdot w$ where $\text{Re}(z \cdot \overline{w})$ gives the projection scalar along $w$.

Reflection $z_r$:

The reflection of $z$ with respect to the line $l_w$ involves projecting $z$ onto the line, then reflecting the imaginary component. In terms of the reflection formula: $z_r = 2z_o - z$ Substitute $z_o$ from the above projection expression: $z_r = 2 \cdot \text{Re}(z \cdot \overline{w}) \cdot w - z$

Summary:

• $z_o = \text{Re}(z \cdot \overline{w}) \cdot w$
• $z_r = 2 \cdot \text{Re}(z \cdot \overline{w}) \cdot w - z$

These expressions use the geometric properties of projection and reflection in the complex plane, leveraging the conjugate $\overline{w}$ to rotate the problem into a simpler form.

Would you like a more detailed breakdown of any step, or do you have any specific questions?

5 Related Questions:

1. How do we geometrically interpret the projection of a complex number onto a line?
2. What is the role of the conjugate $\overline{w}$ in the rotation of the complex plane?
3. How does the reflection formula change if $w$ lies on a different part of the unit circle?
4. Can these projection and reflection formulas be extended to complex numbers with $|w| \neq 1$?
5. What happens to $z_o$ and $z_r$ if $z$ already lies on the line $l_w$?

Tip:

When dealing with transformations in the complex plane, using conjugates is a powerful technique to simplify rotations and projections by aligning points along the real axis.