1. What is Reflection Over Y = X?

Reflection over the line y = x is a geometric transformation that swaps the x and y coordinates of a point. This transformation maps the point (x, y) to the point (y, x), effectively reflecting it across the line where x equals y.

2. How Does the Calculator Work?

The calculator uses the reflection formula:

Where:

• \( x \) — Original x-coordinate
• \( y \) — Original y-coordinate

Explanation: The reflection simply swaps the x and y coordinates, creating a mirror image across the line y = x.

3. Importance of Reflection Calculation

Details: Reflection over y = x is fundamental in coordinate geometry, used in matrix transformations, computer graphics, and understanding symmetry properties in mathematical functions.

4. Using the Calculator

Tips: Enter the x and y coordinates of the point you want to reflect. The calculator will instantly compute and display the reflected coordinates.

5. Frequently Asked Questions (FAQ)

Q1: What happens to points on the line y = x?
A: Points on the line y = x remain unchanged after reflection because their x and y coordinates are already equal.

Q2: How does reflection affect geometric shapes?
A: Reflection over y = x preserves distances and angles but reverses orientation, creating a mirror image of the original shape.

Q3: What is the matrix representation of this reflection?
A: The reflection can be represented by the matrix: \[ \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \]

Q4: How is this reflection used in real-world applications?
A: It's used in computer graphics for mirror effects, in physics for symmetry analysis, and in mathematics for function transformation studies.

Q5: Does this reflection preserve area and perimeter?
A: Yes, reflection is an isometric transformation that preserves distances, angles, areas, and perimeters of geometric figures.