Further Maths/Lessons/Linear Transformations/Successive Transformations & 3D
Core Pure · Linear Transformations · Year 1

Successive Transformations & Three Dimensions

Combining transformations means combining matrices — but the order you multiply in is the order the transformations are applied, read right to left.

13 min read 3 worked examples

Combining transformations

Theorem 2.1 — composition of transformations

If transformation $M_1$ is applied first, followed by $M_2$, the single matrix representing the combined transformation is $M_2M_1$ — not $M_1M_2$. The matrix applied first is written on the right, closest to the coordinate vector.

Exam technique

Read composite transformations right to left: in $M_2M_1\mathbf{x}$, $\mathbf{x}$ is transformed by $M_1$ first, then the result is transformed by $M_2$. Getting this order backwards is one of the most common errors in this topic.

Example 1 · Combining a rotation and reflection
$M_1=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ represents a $90°$ rotation, applied first, followed by $M_2=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$, a reflection in the $x$-axis. Find the single matrix representing the combined transformation.

The combined matrix is $M_2M_1$:

$$M_2M_1=\begin{pmatrix}1&0\\0&-1\end{pmatrix}\begin{pmatrix}0&-1\\1&0\end{pmatrix}=\begin{pmatrix}0&-1\\-1&0\end{pmatrix}$$

This composite matrix represents reflection in the line $y=-x$.

Transformations in three dimensions

Definition 2.2 — 3×3 transformation matrices

A $3\times3$ matrix transforms points in 3D space in the same way: the columns are the images of $\mathbf{i}=(1,0,0)$, $\mathbf{j}=(0,1,0)$, $\mathbf{k}=(0,0,1)$. Rotations about the coordinate axes have standard forms — for example, rotation by $\theta$ about the $z$-axis:

$$\begin{pmatrix}\cos\theta&-\sin\theta&0\\\sin\theta&\cos\theta&0\\0&0&1\end{pmatrix}$$
A 3D coordinate system with x, y and z axes, showing a solid rotating about the z-axis with a curved arrow indicating the direction of rotation in the xy-plane
Fig. 1 — rotation about the z-axis leaves the z-coordinate unchanged and rotates the x–y plane, matching the top-left 2×2 block of the rotation matrix.
Example 2 · A reflection in 3D
Write down the $3\times3$ matrix representing a reflection in the $xy$-plane (i.e. $z\to-z$, $x,y$ unchanged).

$\mathbf{i}\to\mathbf{i}$, $\mathbf{j}\to\mathbf{j}$, $\mathbf{k}\to-\mathbf{k}$, so the columns give:

$$\begin{pmatrix}1&0&0\\0&1&0\\0&0&-1\end{pmatrix}$$
Example 3 · Composing two 3D transformations
Find the image of the point $(1,2,3)$ under a rotation by $90°$ about the $z$-axis, followed by reflection in the $xy$-plane.

Rotation matrix with $\theta=90°$: $R=\begin{pmatrix}0&-1&0\\1&0&0\\0&0&1\end{pmatrix}$. Reflection matrix: $F=\begin{pmatrix}1&0&0\\0&1&0\\0&0&-1\end{pmatrix}$.

Combined matrix (rotation first, so it's on the right): $FR$.

$$FR=\begin{pmatrix}1&0&0\\0&1&0\\0&0&-1\end{pmatrix}\begin{pmatrix}0&-1&0\\1&0&0\\0&0&1\end{pmatrix}=\begin{pmatrix}0&-1&0\\1&0&0\\0&0&-1\end{pmatrix}$$
$$FR\begin{pmatrix}1\\2\\3\end{pmatrix}=\begin{pmatrix}-2\\1\\-3\end{pmatrix}$$

The image is $(-2,1,-3)$.

Common mistake

Writing $M_1M_2$ instead of $M_2M_1$ for "M₁ then M₂" is the single most common error in this topic — since matrix multiplication is not commutative, the two orders generally give completely different transformations.

Exam-style questions

Question 1 6 marks

$M_1$ represents an enlargement scale factor $2$; $M_2=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ represents reflection in $y=x$. Find the single matrix for "enlargement, then reflection" and use it to find the image of $(3,-1)$.

$M_1=\begin{pmatrix}2&0\\0&2\end{pmatrix}$; combined matrix is $M_2M_1$ (M1)

$M_2M_1=\begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}2&0\\0&2\end{pmatrix}=\begin{pmatrix}0&2\\2&0\end{pmatrix}$ (M1, A1)

$\begin{pmatrix}0&2\\2&0\end{pmatrix}\begin{pmatrix}3\\-1\end{pmatrix}=\begin{pmatrix}-2\\6\end{pmatrix}$ (M1, A1)

Image is $(-2,6)$. (A1)

Question 2 5 marks

Write down the $3\times3$ matrix for a rotation of $180°$ about the $x$-axis (leaving $x$ unchanged, and reversing the sign of $y$ and $z$), and find the image of $(2,3,-1)$.

$\begin{pmatrix}1&0&0\\0&-1&0\\0&0&-1\end{pmatrix}$ (M1, A1)

$\begin{pmatrix}1&0&0\\0&-1&0\\0&0&-1\end{pmatrix}\begin{pmatrix}2\\3\\-1\end{pmatrix}=\begin{pmatrix}2\\-3\\1\end{pmatrix}$ (M1, A1)

Image is $(2,-3,1)$. (A1)

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