Geometric Transformations with Matrices (Edexcel A-Level Further Mathematics): Revision Notes

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2.2.2 Geometric Transformations with Matrices

Introduction to 3D Transformations

In 3D geometry, transformations using matrices extend the 2D concepts to three dimensions. These transformations include reflections across planes and rotations about coordinate axes. This topic assumes knowledge of 3D vectors, providing a foundation for visualising and calculating transformations in space.

Key 3D Transformations and Their Matrices

Reflections

A reflexion in 3D mirrors points across a specified plane. The transformation matrices for reflections about the three coordinate planes are:

Reflection in the plane x=0x = 0 (yz−planeyz-plane):

Matrix=(−100010001)\text{Matrix} = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}

Reflection in the plane y=0y = 0 (xz−planexz-plane):

Matrix=(1000−10001)\text{Matrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}

Reflection in the plane z=0z = 0 (xy−planexy-plane):

Matrix=(10001000−1)\text{Matrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix}

Rotations

Rotations in 3D are performed about one of the coordinate axes:

Rotation by angle θ\theta about the x−axisx-axis:

Matrix=(1000cos⁡θ−sin⁡θ0sin⁡θcos⁡θ)\text{Matrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta \\ 0 & \sin \theta & \cos \theta \end{pmatrix}

Rotation by angle θ\theta about the y−axisy-axis:

Matrix=(cos⁡θ0sin⁡θ010−sin⁡θ0cos⁡θ)\text{Matrix} = \begin{pmatrix} \cos \theta & 0 & \sin \theta \\ 0 & 1 & 0 \\ -\sin \theta & 0 & \cos \theta \end{pmatrix}

Rotation by angle θ\theta about the z−axisz-axis:

Matrix=(cos⁡θ−sin⁡θ0sin⁡θcos⁡θ0001)\text{Matrix} = \begin{pmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix}

These matrices rotate vectors counterclockwise when looking along the positive axis direction.

Combined Transformations

When combining multiple transformations, the order of multiplication is critical. The resulting matrix is the product of the matrices of the individual transformations in reverse order:

If transformation AA is applied first, followed by BB, the combined transformation matrix is B×AB \times A

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Example: To apply a rotation about the z−axisz-axis followed by a reflexion in the plane y=0y = 0:

Rotation matrix Rz(θ)R_z(\theta):

Rz(θ)=(cos⁡θ−sin⁡θ0sin⁡θcos⁡θ0001)R_z(\theta) = \begin{pmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix}

Reflexion matrix RefyRef_y:

Refy=(1000−10001)Ref_y = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}

Combined matrix Refy×Rz(θ)Ref_y \times R_z(\theta)

=(1000−10001)= \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}=(cos⁡θ−sin⁡θ0sin⁡θcos⁡θ0001)= \begin{pmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix}=(cos⁡θ−sin⁡θ0−sin⁡θ−cos⁡θ0001)= \begin{pmatrix} \cos \theta & -\sin \theta & 0 \\ -\sin \theta & -\cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix}
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Example: Determine the matrix for a reflexion in the x=0x = 0 plane followed by a rotation of 90∘90^\circ about the z−axisz-axis.

Reflexion matrix:

(−100010001)\begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}

Rotation matrix (90°90° about z−axisz-axis):

(0−10100001)\begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}

Combined transformation:

=(0−10100001)= \begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}=(−100010001)= \begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}=(0−10−100001)= \begin{pmatrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}

Note Summary

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Common Mistakes:

  1. Incorrect Order of Matrix Multiplication: The order in which matrices are multiplied matters. Applying transformations in the wrong sequence gives incorrect results.
  2. Misidentifying Axes for Rotation: Confusing rotations about the x,yx, y, and zz axes.
  3. Incorrect Application of Angles: Using clockwise rotations instead of anticlockwise, or vice versa.
  4. Incorrect Reflexion Plane: Mistaking x=0x = 0 for y=0y = 0, or similar errors.
  5. Determinant Misuse: Misinterpreting the determinant as a scale factor in transformations where it doesn't apply.
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Key Formulas:

Reflection Matrices:

x=0x = 0

(−100010001)\begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}

y=0y = 0

(1000−10001)\begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}

z=0z = 0

(10001000−1)\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix}

Rotation Matrices:

About xx:

(1000cos⁡θ−sin⁡θ0sin⁡θcos⁡θ)\begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta \\ 0 & \sin \theta & \cos \theta \end{pmatrix}

About yy:

(cos⁡θ0sin⁡θ010−sin⁡θ0cos⁡θ)\begin{pmatrix} \cos \theta & 0 & \sin \theta \\ 0 & 1 & 0 \\ -\sin \theta & 0 & \cos \theta \end{pmatrix}

About zz:

(cos⁡θ−sin⁡θ0sin⁡θcos⁡θ0001)\begin{pmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix}

Combined Transformations:

Resulting matrix=B×A\text{Resulting matrix} = B \times A

for applying transformation AA followed by BB.

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