Lesson 11.4: Transformation Matrices & Coordinates Calculations

Learning Objectives

• Understand how transformation matrices work for 2D coordinates.
• Apply matrices to perform translations, rotations, reflections, and scaling.
• Calculate new coordinates after transformations.

Topics Covered

• Translation matrices
• Rotation matrices
• Reflection matrices
• Scaling matrices
• Matrix multiplication to transform coordinates
• Combined transformations

Key Terms

• Transformation Matrix: A matrix used to apply a geometric transformation to coordinates.
• Translation: Shifting a point or shape by a certain distance.
• Rotation: Turning a point or shape around the origin or a fixed point.
• Reflection: Flipping a shape across a line.
• Scaling: Enlarging or shrinking a shape by a scale factor.

Lesson Examples

Example 1: Translation

Translate point \( P(2, 3) \) by vector \( \vec{v} = (4, -2) \).

New coordinates: \( P' = (2+4, 3-2) = (6, 1) \).

Example 2: Rotation 90° about origin

Rotate point \( P(1, 2) \) 90° counterclockwise:

Rotation matrix: \(\begin{bmatrix}0 & -1\\1 & 0\end{bmatrix}\), so \( P' = \begin{bmatrix}0 & -1\\1 & 0\end{bmatrix}\begin{bmatrix}1\\2\end{bmatrix} = \begin{bmatrix}-2\\1\end{bmatrix} \).

Example 3: Reflection across x-axis

Reflect \( P(3, -4) \) across the x-axis:

Reflection matrix: \(\begin{bmatrix}1 & 0\\0 & -1\end{bmatrix}\), so \( P' = (3, 4) \).

Example 4: Scaling

Scale \( P(2, 5) \) by factor 3:

Scaling matrix: \(\begin{bmatrix}3 & 0\\0 & 3\end{bmatrix}\), so \( P' = (6, 15) \).

Practice Problems

1. Translate \( A(1,1) \) by vector \( (3,4) \).
2. Rotate \( B(2,3) \) 180° about the origin.
3. Reflect \( C(-2,5) \) across the y-axis.
4. Scale \( D(1,2) \) by factors 2 (x-axis) and 3 (y-axis).
5. Combine translation by (2, -1) and rotation 90° of \( E(1,0) \).

Tips & Common Mistakes

• Always write points as column vectors before multiplying by a matrix.
• Check the order of combined transformations—matrix multiplication is not commutative.
• Double-check signs for reflections and rotations.

Summary

Transformation matrices allow precise calculation of new coordinates under translations, rotations, reflections, and scaling. Combined transformations are handled by matrix multiplication.

Challenge Problems

1. Rotate \( F(3,4) \) 270° counterclockwise and then translate by (-2,1).
2. Reflect \( G(5,-3) \) across the line y=x.
3. Scale \( H(-1,2) \) by 0.5 in x-direction and 2 in y-direction, then rotate 90° CCW.