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解释一下为什么cosβ是这样算的

To explain why \(\cos \beta\) is calculated in this way, let's break it down step by step:

1. **Geometry Setup**: 
   The diagram shows a light path between points \( P_1 \), \( P_2 \), and the final point \( E \). The angle \( \beta \) is the angle between the line segment \( P_1P_2 \) and the normal vector \( \mathbf{n} \) to a surface.

2. **Formula Explanation**: 
   The formula for the cosine of the angle between two vectors is given by:
   \[
   \cos \beta = \frac{\mathbf{P_1P_2} \cdot \mathbf{n}}{|\mathbf{P_1P_2}| |\mathbf{n}|}
   \]
   This is a general formula for the cosine of the angle between two vectors, derived from the dot product. The dot product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is defined as:
   \[
   \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos \theta
   \]
   where \( \theta \) is the angle between the vectors \( \mathbf{A} \) and \( \mathbf{B} \). Rearranging this gives the formula for \( \cos \theta \).

3. **Vectors Definition**:
   - The vector \( \mathbf{P_1P_2} \) represents the direction from point \( P_1 \) to point \( P_2 \), and its coordinates are given by:
     \[
     \mathbf{P_1P_2} = (r_2 - r_1, -z_1)
     \]
     This vector is determined by the change in the radial and vertical (z-axis) positions between points \( P_1 \) and \( P_2 \).
   - The normal vector \( \mathbf{n} \) is a unit vector in the radial direction, perpendicular to the surface at \( P_2 \). Here, it is defined as:
     \[
     \mathbf{n} = (0, 1)
     \]
     meaning that the normal is pointing vertically upwards along the z-axis.

4. **Dot Product Calculation**:
   The dot product \( \mathbf{P_1P_2} \cdot \mathbf{n} \) is calculated as:
   \[
   \mathbf{P_1P_2} \cdot \mathbf{n} = (r_2 - r_1) \cdot 0 + (-z_1) \cdot 1 = -z_1
   \]
   This shows that the dot product depends only on the vertical component \( -z_1 \), meaning the radial change \( r_2 - r_1 \) does not contribute to the dot product because the normal vector has no radial component.

5. **Magnitude of the Vectors**:
   - The magnitude of \( \mathbf{P_1P_2} \) is:
     \[
     |\mathbf{P_1P_2}| = \sqrt{(r_2 - r_1)^2 + (-z_1)^2}
     \]
   - The magnitude of \( \mathbf{n} \) is simply 1, since it is a unit vector:
     \[
     |\mathbf{n}| = 1
     \]

6. **Final Expression for \( \cos \beta \)**:
   Substituting these values into the formula for \( \cos \beta \):
   \[
   \cos \beta = \frac{-z_1}{\sqrt{(r_2 - r_1)^2 + (-z_1)^2}}
   \]

This shows that the angle \( \beta \) is determined by the relationship between the vertical distance \( z_1 \) and the total distance between \( P_1 \) and \( P_2 \), which includes both the radial and vertical changes.

### Would you like further details or have any specific questions?

Here are 5 related questions to expand this topic:
1. How do we calculate the magnitude of vectors in 3D space?
2. What is the geometric interpretation of the dot product in this context?
3. How would the calculation change if the normal vector \( \mathbf{n} \) had a radial component?
4. Can you explain the significance of the radial distance \( r_2 - r_1 \) in the vector \( \mathbf{P_1P_2} \)?
5. How does the unit normal vector simplify the calculation of \( \beta \)?

**Tip**: Always ensure that vectors are correctly defined with their components before applying the dot product or cross product operations.

Understand how to calculate cosβ in a vector dot product scenario using the light path between points P1, P2, and a normal vector n. This problem involves calculating the angle between vectors P1P2 and n using trigonometry and vector algebra. Suitable for undergraduate mathematics or physics students.

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