Lesson 7 The Rotation Matrix

7.2 Lecture Notes

7.2.1 Rotation Matrix in \(\mathbb{R}^2\)

The rotation matrix in \(\mathbb{R}^2\) for a counterclockwise rotation by an angle \(\theta\) is:

\[ R(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \]

  • When this matrix is applied to a vector \(\mathbf{v} \in \mathbb{R}^2\), it rotates \(\mathbf{v}\) counterclockwise by \(\theta\) radians about the origin.

7.2.2 Geometric Interpretation: Left Multiplication as Rotation

Multiplying a 2D vector \(\mathbf{v}\) on the left by \(R(\theta)\):

\[ R(\theta) \cdot \mathbf{v} \]

geometrically rotates the vector \(\mathbf{v}\) in the plane, without changing its length (if \(\mathbf{v} \neq \mathbf{0}\)).

  • The direction changes, but the magnitude remains the same.
  • This is an isometry: a transformation that preserves distances and angles.

7.2.3 Stretching and Compression

A diagonal matrix can stretch or compress a vector along the coordinate axes.

For example:

\[ S = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix} \]

  • This stretches vectors by a factor of 2 in the \(x\)-direction and leaves the \(y\)-direction unchanged.

7.2.4 Reflection Matrices

Common reflection matrices in \(\mathbb{R}^2\):

  • Reflection across the \(x\)-axis:

\[ \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \]

  • Reflection across the \(y\)-axis:

\[ \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} \]

  • Reflection across the line \(y = x\):

\[ \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \]

  • These matrices flip vectors across the specified axis or line, changing direction while preserving length.

7.2.5 Summary

Transformation Type Matrix Form Effect
Rotation \(R(\theta)\) Turns a vector counterclockwise
Stretch/Compression Diagonal Scales components independently
Reflection Specific matrices Flips vectors across lines or axes

7.3 Practice Problems

  1. Rotate
    \[ \begin{bmatrix}1 \\ -2 \end{bmatrix} \]
    60 degrees counter clockwise.
  2. Rotate
    \[ \begin{bmatrix}1 \\ 1 \end{bmatrix} \]
    90 degrees counter clockwise.
  3. Multiply, then plot the original vector and the rotated vector:
    \[ \begin{bmatrix} \frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & \frac{1}{2} \end{bmatrix} \begin{bmatrix}1 \\ -2 \end{bmatrix}. \]
  4. Plot the original vector. Then multiply the vector one matrix at a time, plotting all output vectors along the way. Describe what each matrix multiplication does to the vector:
    \[ \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} \]
  5. Applied: Let \(B = \left\{ \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \begin{bmatrix} -1 \\ 1 \end{bmatrix} \right\}\). Let \(s_B = \begin{bmatrix} 3 \\ -2 \end{bmatrix}\). Express \(s_B\) in terms of the standard basis, rotate it 30 degrees counter clockwise, then express that vector in terms of basis \(B\).

7.4 Self-Assessment

Time yourself and try to solve the following questions within twenty minutes.

  1. Rotate \(\begin{bmatrix}-1 \\ -2 \end{bmatrix}\) 30 degrees clockwise.

  2. Multiply, then plot the original vector and the rotated vector:

    \[ \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix}1 \\ -2 \end{bmatrix} \]

  3. Explain what left-sided multiplication by
    \(A=\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}\)
    would do to the vector \(\begin{bmatrix}x \\ y \end{bmatrix}\).
    Are there any vectors in \(\mathbb{R}^2\) that would not change after left-sided multiplication by \(A\)?

  4. Let \(B = \left\{ \begin{bmatrix} -1 \\ -1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right\}\).
    Let \(s_B = \begin{bmatrix} -1 \\ -3 \end{bmatrix}\).
    Express \(s_B\) in terms of the standard basis, rotate it 60 degrees counter clockwise, then express that vector in terms of basis \(B\).

7.5 Lesson Checklist

This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an \(X\) in the appropriate box beside the skill below.

\[\begin{align*} &\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\ &\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\ &\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.} \end{align*}\]

Skill D CON COM
Find and simplify a rotation matrix.
Plot results of a rotation geometrically.
Solve an applied problem involving change of basis and rotations.