Linear Transformations in 3D

This lesson extends linear transformations to three dimensions. We use 3×3 matrices to represent rotations, reflections, and other transformations in 3D space. The principles are the same as in 2D, but the geometry is richer.

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3D Transformation Matrices

A linear transformation $T: \mathbb{R}^3 \to \mathbb{R}^3$T:R3→R3 is represented by a 3×3 matrix $M$M. The columns of $M$M are the images of the standard basis vectors:

$M = \begin{pmatrix} | & | & | \\ T(\mathbf{e}_1) & T(\mathbf{e}_2) & T(\mathbf{e}_3) \\ | & | & | \end{pmatrix}$M=​∣T(e1​)∣​∣T(e2​)∣​∣T(e3​)∣​​

where $\mathbf{e}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$e1​=​100​​, $\mathbf{e}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$e2​=​010​​, $\mathbf{e}_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$e3​=​001​​.

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Standard 3D Transformations

Reflections

Reflection in plane	Matrix
$xy$xy-plane ($z = 0$z=0)	$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix}$​100​010​00−1​​
$xz$xz-plane ($y = 0$y=0)	$\begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$​100​0−10​001​​
$yz$yz-plane ($x = 0$x=0)	$\begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$​−100​010​001​​

Rotations About Coordinate Axes

Rotation by $\theta$θ about the z-axis:

$$R_z(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix}$$Rz​(θ)=​cosθsinθ0​−sinθcosθ0​001​​

The z-component is unchanged; the x and y components rotate as in 2D.

Rotation by $\theta$θ about the x-axis:

$$R_x(\theta) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{pmatrix}$$Rx​(θ)=​100​0cosθsinθ​0−sinθcosθ​​

Rotation by $\theta$θ about the y-axis:

$$R_y(\theta) = \begin{pmatrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \end{pmatrix}$$Ry​(θ)=​cosθ0−sinθ​010​sinθ0cosθ​​

Other Transformations

Transformation	Matrix
Enlargement, factor $k$k	$kI = \begin{pmatrix} k & 0 & 0 \\ 0 & k & 0 \\ 0 & 0 & k \end{pmatrix}$kI=​k00​0k0​00k​​
Stretch in z-direction, factor $c$c	$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & c \end{pmatrix}$​100​010​00c​​
Projection onto the $xy$xy-plane	$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}$​100​010​000​​

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Worked Examples

Example 1: Find the image of the point $(1, 2, 3)$(1,2,3) under reflection in the $xz$xz-plane.

$$\begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix}$$​100​0−10​001​​​123​​=​1−23​​