Reflecting Light Through a Coordinate System: A Detailed Mathematical Analysis

Introduction

This article delves into the nuances of reflecting light through a coordinate system, specifically focusing on the geometric principles involved in determining the coordinates of a point of reflection. The problem at hand involves a ray of light passing through a given point, reflecting off the x-axis, and then passing through another point. We will use the principles of geometry and trigonometry to solve this problem.

The Problem

Consider a ray of light passing through the point (1, 2) and reflecting off the x-axis at a point A, such that the reflected ray passes through the point (5, 3). The goal is to determine the coordinates of point A.

Solution

Step 1: Setting Up the Coordinates

Let the point of incidence be A with coordinates (1, 2), and let the point of reflection be E with unknown coordinates (E, 0).

Step 2: Using Angle of Incidence and Reflection

Since the law of reflection states that the angle of incidence is equal to the angle of reflection, we can set up the angles based on the normal to the x-axis.

The angle between the x-axis and the incident ray is π - θ. The angle between the x-axis and the reflected ray is θ.

The slopes of the incident and reflected rays can be derived as follows:

The slope of the incident ray (from (1, 2) to (E, 0)) is:

tan(π - θ) -tan(θ)

The slope of the reflected ray (from (E, 0) to (5, 3)) is:

tan(θ)

Step 3: Equations of the Rays

Using the point-slope form of the line equation, we can write:

For the incident ray (through (1, 2) and (E, 0)):

y - 2 m_1 (x - 1)

For the reflected ray (through (E, 0) and (5, 3)):

y m_2 (x - E)

Since m_1 -tan(θ) and m_2 tan(θ), we can write the equations as:

2 -tan(θ) (1 - E)

3 tan(θ) (5 - E)

Step 4: Solving for E

Let's solve these equations to find the value of E.

From the first equation:

2 -tan(θ) (1 - E)

2 tan(θ) (E - 1)

E - 1 2 cot(θ)

E 2 cot(θ) 1 ---(1)

From the second equation:

3 tan(θ) (5 - E)

3 tan(θ) (5 - E)

5 - E 3 cot(θ)

E 5 - 3 cot(θ) ---(2)

Multiplying equation (1) by 3 and equation (2) by 2, we get:

6 cot(θ) 3 3E

10 - 6 cot(θ) 2E

Adding these two equations:

13 5E

E 13/5

Hence, the coordinates of point A are (13/5, 0).

Conclusion

This detailed analysis demonstrates the application of geometric principles and trigonometry in solving problems related to the reflection of light. By using the law of reflection and the point-slope form of the line equation, we were able to determine the exact coordinates of the point of reflection, providing a clear and logical solution to the problem.

Key Takeaways

The angle of incidence is equal to the angle of reflection. Slopes of incident and reflected rays can be used to set up equations for the rays. Using the point-slope form and solving simultaneous equations, the coordinates of the point of reflection can be found.

Further Reading

Geometry of Reflection Solving Systems of Equations Trigonometry Basics