∣*z*∣∣*z*0​∣+∣*z*0​∣∣*z*∣=(∣*z*∣+∣*z*0​∣)⋅∣*z*−*z*0​∣ how is this true for complex numbers



The equation

\[
|z||z_0| + |z_0||z| = (|z| + |z_0|) \cdot |z - z_0|
\]

can be rewritten more clearly as:

\[
2 |z||z_0| = (|z| + |z_0|) \cdot |z - z_0|
\]

This equation holds for complex numbers under certain conditions. To understand why, let's break it down.

### Step 1: Consider the properties of the modulus of complex numbers

For any complex numbers \( z \) and \( z_0 \):

- The modulus \( |z| \) of a complex number \( z = x + iy \) (where \( x, y \) are real numbers and \( i \) is the imaginary unit) is defined as \( |z| = \sqrt{x^2 + y^2} \).
- The modulus function satisfies the **triangle inequality**, which states that \( |z + w| \leq |z| + |w| \) for any two complex numbers \( z \) and \( w \).
- The modulus also satisfies the **equality condition**: \( |z - z_0| = |z| - |z_0| \) if and only if \( z \) and \( z_0 \) are real and have the same argument, meaning they are on the same line through the origin in the complex plane.

### Step 2: Analyze the equation

Now, let's consider the equation \( 2|z||z_0| = (|z| + |z_0|) \cdot |z - z_0| \):

1. **Left-hand side:** \( 2|z||z_0| \) is simply twice the product of the magnitudes of \( z \) and \( z_0 \).

2. **Right-hand side:** \( (|z| + |z_0|) \cdot |z - z_0| \) represents the product of the sum of the magnitudes and the distance between \( z \) and \( z_0 \).

### Step 3: Condition for equality

The equality \( 2|z||z_0| = (|z| + |z_0|) \cdot |z - z_0| \) holds **if and only if** the complex numbers \( z \) and \( z_0 \) satisfy specific geometric conditions. Specifically:

- **Collinearity:** \( z \) and \( z_0 \) must lie on the same line through the origin in the complex plane. This means that \( z = \lambda z_0 \) for some real number \( \lambda \).
- **Specific values:** If \( z_0 = 0 \) or \( z = z_0 \), the equation is trivially true. 

Given the above conditions, the equality simplifies to both sides being equal due to the linear relationship between \( z \) and \( z_0 \).

### Conclusion

The equation is true for complex numbers when \( z \) and \( z_0 \) are collinear and have the same direction (i.e., they lie on the same ray from the origin).

Would you like more details or have any further questions? Here are some related questions:

1. What is the triangle inequality for complex numbers?
2. How does the argument of a complex number affect its modulus?
3. Can you show examples of when this equation holds true for specific complex numbers?
4. What happens if \( z \) and \( z_0 \) are orthogonal in the complex plane?
5. How can this equation be used in practical applications involving complex numbers?

**Tip:** For any equation involving complex numbers, visualizing the points on the complex plane often helps to understand their relationships better.

Explore why the equation |z||z_0| + |z_0||z| = (|z| + |z_0|) \cdot |z - z_0| holds true for complex numbers. Discover the geometric conditions under which this equation is satisfied, with detailed explanations and examples. Suitable for advanced high school and college-level students.

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