Compose two plane transformations
Read the transformations in the order in which they act on the point. Multiply matrix parts in reverse writing order and calculate the translation vector separately
\[M_{\mathrm{comp}}=M_2M_1,\quad\mathbf{t}_{\mathrm{comp}}=M_2\mathbf{t}_1+\mathbf{t}_2\]
- Read the chronology The first named transformation acts on the original point before the second transformation acts on the result \(\displaystyle \mathbf{p}\xrightarrow{T_1}\mathbf{p}_1\xrightarrow{T_2}\mathbf{p}_2\)
- Multiply matrices in acting order The matrix for the second transformation is written on the left of the matrix for the first transformation \(\displaystyle M_{\mathrm{comp}}=M_2M_1\)
- Transform translation vectors The second matrix acts on the first translation vector before the second translation vector is added \(\displaystyle \mathbf{t}_{\mathrm{comp}}=M_2\mathbf{t}_1+\mathbf{t}_2\)
- Compare the reverse order For a non-commuting pair, reversing the transformations can change the composite matrix, translation vector or image point \(\displaystyle T_2(T_1(\mathbf{p}))\ne T_1(T_2(\mathbf{p}))\)
Worked example
The point shown below is reflected in the x-axis and then dilated from the y-axis by factor 2. Find its image point
\[\mathbf{p}=\begin{pmatrix}-2\\5\end{pmatrix},\quad M_1=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\quad M_2=\begin{pmatrix}2&0\\0&1\end{pmatrix}\]
- \(\displaystyle M_1\mathbf{p}=\begin{pmatrix}1&0\\0&-1\end{pmatrix}\begin{pmatrix}-2\\5\end{pmatrix}=\begin{pmatrix}-2\\-5\end{pmatrix}\)The reflection is the first step, so it acts directly on the original point and changes the y-coordinate sign
- \(\displaystyle M_2(M_1\mathbf{p})=\begin{pmatrix}2&0\\0&1\end{pmatrix}\begin{pmatrix}-2\\-5\end{pmatrix}=\begin{pmatrix}-4\\-5\end{pmatrix}\)The dilation from the y-axis multiplies the x-coordinate by 2 and leaves the y-coordinate unchanged
- \(\displaystyle M_2M_1=\begin{pmatrix}2&0\\0&-1\end{pmatrix}\)The composite matrix is the second matrix on the left of the first matrix
- \(\displaystyle \mathbf{p}'=M_2M_1\mathbf{p}=\begin{pmatrix}-4\\-5\end{pmatrix}\)The final image point is obtained after both transformations have acted in order
Common mistakes
Reversing the composition order
✕\(\displaystyle M_{\mathrm{comp}}=M_1M_2\)✓\(\displaystyle M_{\mathrm{comp}}=M_2M_1\)
The first transformation acts nearest to the original point, so its matrix is on the right of the second matrixAdding translation vectors without transforming the first
✕\(\displaystyle \mathbf{t}_{\mathrm{comp}}=\mathbf{t}_1+\mathbf{t}_2\)✓\(\displaystyle \mathbf{t}_{\mathrm{comp}}=M_2\mathbf{t}_1+\mathbf{t}_2\)
The second transformation acts on the first translation vector before the second translation vector is addedOmitting a translation vector
✕\(\displaystyle \mathbf{p}'=M_2M_1\mathbf{p}\)✓\(\displaystyle \mathbf{p}'=M_2M_1\mathbf{p}+M_2\mathbf{t}_1+\mathbf{t}_2\)
A translation is kept as a vector and is transformed or added separately from the matrix partsDifficulty levels
introCompose two origin-fixing matrix transformations in the stated order
standardCompose a reflection and a dilation and follow their chronological order
challengeCompose one translation and one matrix transformation and compare the reverse order
examCompare orders, construct the composite matrix and vector, and apply them to a point
Questions about this topic
Why is the first matrix written on the right?+
The first transformation acts first on the point, so the second transformation's matrix is applied to the result and is written on the left
How is the translation part of a composite found?+
Apply the second matrix to the first translation vector, then add the second translation vector
Do two transformations always give the same result in either order?+
No. Some pairs commute, but many pairs have different composite matrices or translation vectors when their order is reversed
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 7: Matrices and plane transformations
In this chapter
Connect a transformed rule, graph and coordinate mapWrite simultaneous equations as a matrix equationSolve simultaneous equations using the inverse matrixSolve a pair of simultaneous linear equationsMap or recover points and finite coordinate setsIdentify a plane transformation and its exact parametersState the order of a matrix and locate its elementsAdd and subtract matricesMultiply a matrix by a scalarCombine scalar multiples of matricesDecide whether a matrix product is defined and give its orderMultiply matricesIdentify and construct the identity matrixFind the determinant of a 2 × 2 matrixFind the inverse of a 2 × 2 matrixCompose two plane transformations in orderEvaluate powers of a square matrixDetermine the image rule of a basic function or relation