Rotation matrices describe rotations about the origin.

The angle of rotation \(\theta\) is counterclockwise off of the positive x-axis (due east). It follows the same convention as the unit circle and the direction of vectors.

Rotation matrices are square matrices, orthogonal matrices and have a determinant of 1.

 

Notes

 

Rotation Matrix
\(R(\theta)=\left[{\begin{array}{cc}\cos{\theta} & -\sin{\theta} \\\sin{\theta} & \cos{\theta} \\\end{array} } \right]\)

 

 

To rotate \(\left[{\begin{array}{c}x \\y\\\end{array} } \right]\) by \(\theta\) degrees
\( \left[{\begin{array}{cc}\cos{\theta} & -\sin{\theta} \\\sin{\theta} & \cos{\theta} \\\end{array} } \right] \left[{\begin{array}{c}x \\y\\\end{array} } \right] = \left[{\begin{array}{c}x’ \\y’\\\end{array} } \right] \)

 

Practice Problems

\(\textbf{1)}\) \( \text{Find the rotation matrix for } 30^{\circ} \)

Show Rotation Matrix

The rotation matrix is \(\left[{\begin{array}{cc}\frac{\sqrt{3}}{2} & -\frac{1}{2} \\\ \frac{1}{2} & \frac{\sqrt{3}}{2} \\\end{array} } \right]\)

 

\(\textbf{2)}\) \( \text{Find the rotation matrix for } 45^{\circ} \)

Show Rotation Matrix

The rotation matrix is \(\left[{\begin{array}{cc}\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\\end{array} } \right]\)

 

\(\textbf{3)}\) \( \text{Find the rotation matrix for } 90^{\circ} \)

Show Rotation Matrix

The rotation matrix is \(\left[{\begin{array}{cc}0 & -1 \\\ 1 & 0 \\\end{array} } \right]\)

 

\(\textbf{4)}\) \( \text{Find the rotation matrix for } 135^{\circ} \)

Show Rotation Matrix

The rotation matrix is \(\left[{\begin{array}{cc}-\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\\ \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\\end{array} } \right]\)

 

\(\textbf{5)}\) \( \text{Find the rotation matrix for } 300^{\circ} \)

Show Rotation Matrix

The rotation matrix is \(\left[{\begin{array}{cc}\frac{1}{2} & \frac{\sqrt{3}}{2} \\\ -\frac{\sqrt{3}}{2} & \frac{1}{2} \\\end{array} } \right]\)

 

Perform the stated rotation

\(\textbf{6)}\) \( \text{Rotate the vector }\left[{\begin{array}{c}1 \\1\\\end{array} } \right] \text{ by } 30^{\circ} \)

Show Answer

The answer is \(\left[{\begin{array}{c}\frac{-1+\sqrt{3}}{2} \\\frac{1+\sqrt{3}}{2}\\\end{array} } \right] \approx \left[{\begin{array}{c}.366 \\1.366\\\end{array} } \right]\)

Show Work

\(\textbf{Step 1: Rotation Matrix Notes}\)
\(\,\,\,\,\, \left[{\begin{array}{cc}\cos{\theta} & -\sin{\theta} \\\sin{\theta} & \cos{\theta} \\\end{array} } \right] \left[{\begin{array}{c}x \\y\\\end{array} } \right] \)
\(\,\,\,\,\, \left[{\begin{array}{cc}\cos{30} & -\sin{30} \\\sin{30} & \cos{30} \\\end{array} } \right] \left[{\begin{array}{c}1 \\1\\\end{array} } \right]\)
\(\,\,\,\,\, \left[{\begin{array}{cc}\frac{\sqrt{3}}{2} & -\frac{1}{2} \\\ \frac{1}{2} & \frac{\sqrt{3}}{2} \\\end{array} } \right] \left[{\begin{array}{c}1 \\1\\\end{array} } \right]\)
\(\textbf{Step 2: Matrix Multiplication}\)
\(\,\,\,\,\, \left[{\begin{array}{c}\frac{\sqrt{3}}{2}(1) -\frac{1}{2}(1) \\\ \frac{1}{2}(1) + \frac{\sqrt{3}}{2}(1) \\\end{array} } \right] \)
\(\left[{\begin{array}{c}\frac{-1+\sqrt{3}}{2} \\\frac{1+\sqrt{3}}{2}\\\end{array} } \right] \approx \left[{\begin{array}{c}.366 \\1.366\\\end{array} } \right]\)

 

\(\textbf{7)}\) \( \text{Rotate the vector }\left[{\begin{array}{c}5 \\5\\\end{array} } \right] \text{ by } 30^{\circ} \)

Show Answer

The answer is \(\left[{\begin{array}{c}\frac{-5+5\sqrt{3}}{2} \\\frac{5+5\sqrt{3}}{2}\\\end{array} } \right] \approx \left[{\begin{array}{c}1.83 \\6.83\\\end{array} } \right]\)

Show Work

\(\,\,\,\,\, \left[{\begin{array}{cc}\cos{\theta} & -\sin{\theta} \\\sin{\theta} & \cos{\theta} \\\end{array} } \right] \left[{\begin{array}{c}x \\y\\\end{array} } \right] \)
\(\,\,\,\,\, \left[{\begin{array}{cc}\cos{30} & -\sin{30} \\\sin{30} & \cos{30} \\\end{array} } \right] \left[{\begin{array}{c}5 \\5\\\end{array} } \right]\)
\(\,\,\,\,\, \left[{\begin{array}{cc}\frac{\sqrt{3}}{2} & -\frac{1}{2} \\\ \frac{1}{2} & \frac{\sqrt{3}}{2} \\\end{array} } \right] \left[{\begin{array}{c}5 \\5\\\end{array} } \right]\)
\(\,\,\,\,\, \left[{\begin{array}{c}\frac{\sqrt{3}}{2}(5) -\frac{1}{2}(5) \\\ \frac{1}{2}(5) + \frac{\sqrt{3}}{2}(5) \\\end{array} } \right] \)
\(\left[{\begin{array}{c}\frac{-5+5\sqrt{3}}{2} \\\frac{5+5\sqrt{3}}{2}\\\end{array} } \right] \approx \left[{\begin{array}{c}1.83 \\6.83\\\end{array} } \right]\)

 

\(\textbf{8)}\) \( \text{Rotate the vector }\left[{\begin{array}{c}5 \\5\\\end{array} } \right] \text{ by } 45^{\circ} \)

Show Answer

The answer is \(\left[{\begin{array}{c}0 \\5\sqrt{2}\\\end{array} } \right] \approx \left[{\begin{array}{c}0 \\7.07\\\end{array} } \right]\)

 

\(\textbf{9)}\) \( \text{Rotate the vector }\left[{\begin{array}{c}1 \\-4\\\end{array} } \right] \text{ by } 45^{\circ} \)

Show Answer

The answer is \(\left[{\begin{array}{c}\frac{5\sqrt{2}}{2} \\\frac{-3\sqrt{2}}{2}\\\end{array} } \right] \approx \left[{\begin{array}{c}3.54 \\-2.12\\\end{array} } \right]\)

 

\(\textbf{10)}\) \( \text{Rotate the vector }\left[{\begin{array}{c}2 \\3\\\end{array} } \right] \text{ by } 90^{\circ} \)

Show Answer

The answer is \(\left[{\begin{array}{c}-3 \\2\\\end{array} } \right]\)

 

\(\textbf{11)}\) \( \text{Rotate the vector }\left[{\begin{array}{c}-8 \\3\\\end{array} } \right] \text{ by } 135^{\circ} \)

Show Answer

The answer is \(\left[{\begin{array}{c}\frac{5\sqrt{2}}{2} \\\frac{-11\sqrt{2}}{2}\\\end{array} } \right] \approx \left[{\begin{array}{c}3.54 \\-7.78\\\end{array} } \right]\)

 

\(\textbf{12)}\) \( \text{Rotate the vector }\left[{\begin{array}{c}-5 \\-5\\\end{array} } \right] \text{ by } 300^{\circ} \)

Show Answer

The answer is \(\left[{\begin{array}{c}\frac{-5-5\sqrt{3}}{2} \\\frac{-5+5\sqrt{3}}{2}\\\end{array} } \right]\approx \left[{\begin{array}{c}-6.83 \\1.83\\\end{array} } \right]\)

 

 

See Related Pages\(\)

\(\bullet\text{ Intro to Matrices}\)
\(\,\,\,\,\,\,\,\,\)\(\left[ {\begin{array}{ccc}4 & -5 & 2 \\1 & 0 & 3 \\\end{array} } \right]\)

\(\bullet\text{ Matrix Operations}\)
\(\,\,\,\,\,\,\,\,\)\( \left[ {\begin{array}{ccc}3 & 45 & 6 \\-8 & 2 & 4 \\1 & 0 & 3 \\\end{array} } \right]\)\(+\left[ {\begin{array}{ccc}3 & 45 & 6 \\-8 & 2 & 4 \\1 & 0 & 3 \\\end{array} } \right]\)

\(\bullet\text{ Multiplying Matrices}\)
\(\,\,\,\,\,\,\,\,\)\(\left[{\begin{array}{cc} 1 & 2 \\ -3 & -4 \\\end{array} } \right]\)\(\left[ {\begin{array}{cc}6 & -3 \\5 & 0 \\\end{array} } \right]\)

\(\bullet\text{ Determinants}\)
\(\,\,\,\,\,\,\,\,\)\(\left|{\begin{array}{cc} a & b \\ c & d \\ \end{array} } \right|=ad-bc\)

\(\bullet\text{ Cramer’s Rule}\)
\(\,\,\,\,\,\,\,\,\text{ax+by=e } \& \text{ cx+dy=f}…\)

\(\bullet\text{ Identity Matrix}\)
\(\,\,\,\,\,\,\,\,\)\(\left[{\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right]\)

\(\bullet\text{ Identity and Inverse Matrices}\)
\(\,\,\,\,\,\,\,\,A^{-1}=\displaystyle\frac{1}{ad-bc}\left[{\begin{array}{cc} a & b \\ c & d \\ \end{array} } \right]\)

\(\bullet\text{ Transpose Matrix}\)
\(\,\,\,\,\,\,\,\,\left[{\begin{array}{ccc} 1 \\ 2 \\ 5 \\ \end{array} } \right]\Rightarrow\left[{\begin{array}{c} 1 & 2 & 5 \end{array} } \right]\)

\(\bullet\text{ Rotation Matrix}\)
\(\,\,\,\,\,\,\,\,\)\(R(\theta)=\left[{\begin{array}{cc}\cos{\theta} & -\sin{\theta} \\\sin{\theta} & \cos{\theta} \\\end{array} } \right]\)

\(\bullet\text{ Eigenvectors and Eigenvalues}\)
\(\,\,\,\,\,\,\,\,(A-\lambda I)\vec{v}=\vec{0}\)

\(\bullet\text{ Andymath Homepage}\)