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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.

Let w be a complex number such that |w| = 1. Consider the line lw = {aw : a ∈ R} through the origin in the direction of w in the complex plane. For a complex number z, let zo be the orthogonal projection of z onto lw, and let zr be the reflection of z with respect to lw. Express zo and zr in terms of z and w using complex addition, multiplication, and conjugation.

Explore the orthogonal projection and reflection of a complex number onto a line in the direction of w, where |w| = 1. Learn how to express these projections using complex addition, multiplication, and conjugation, covering advanced concepts in the complex plane.

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