REFLECTION MATRIX
Definition Of Reflection Matrix
A matrix that is used to reflect an object over a line or plane is called a reflection matrix.
Examples of Reflection Matrix
The figure below shows the reflection of triangle ABC about the y-axis.
is the reflection matrix for the y-axis.
Video Examples: Reflection Matrix
Solved Example on Reflection Matrix
Ques: Find the coordinates of the vertices of the ../image_md of triangle ABC with A(1, 3), B(-1, 1), C(3, 1) after a reflection across the y-axis.
Choices:
A. A'(- 1, 3), B'(1, 1), C'(- 3, 1)
B. A'(- 1, - 3), B'(1, 1), C'(- 3, 1)
C. A'(- 1, 3), B'(1, - 1), C'(- 3, 1)
D. A'(- 1, 3), B'(1, 1), C'(3, 1)
Solution:
Step 1: The coordinates of the vertices of triangle ABC are given by A(1, 3), B(-1, 1), C(3, 1)
Step 2: Then multiply the vertex matrix by reflection matrix for the y-axis
Step 3: Therefore, the coordinates of the vertices of A'B'C' are A'(- 1, 3), B'(1, 1), C'(- 3, 1).
Quick Summary
- Reflection matrices are used to mirror objects across a line or plane.
- The reflection matrix for the y-axis in 2D is [[-1, 0], [0, 1]].
- Applying a reflection matrix involves matrix multiplication with the coordinate matrix of the object.
🍎 Teacher Insights
Use visual aids and interactive software to demonstrate the effect of reflection matrices on geometric shapes. Emphasize the geometric interpretation of matrix multiplication.🎓 Prerequisites
- Matrix Multiplication
- Coordinate Geometry
- Transformations
- Vectors
Check Your Knowledge
Q1: What is the reflection of the point (2,3) across the y-axis?
Q2: Which matrix represents reflection across the x-axis?
Frequently Asked Questions
Q: How do I find the reflection matrix for a line other than the x or y axis?
A: You can use a rotation matrix to align the line with the x-axis, perform the reflection, and then rotate back.
Q: Is a reflection matrix orthogonal?
A: Yes, reflection matrices are orthogonal, meaning their transpose is equal to their inverse.