Order of composition when dealing with transformations

Question

I have been struggling with a question in my book.

• $T$ is a translation of $(+5,+4)$,
• $M$ is a reflection in the line $y=x$.
• $R$ is a 90 degree anticlockwise rotation about $(0,0)$

Write down the coordinates of the image of the point (7,9) under the following transformations:

• TR - to get the correct answer, I had to rotate first then translate.
• RT - to get the correct answer, I had to translate first then rotate.

Did the guys who made the answer key get mixed up?

Another doubt: When asked to feed (7,9) into transformation TM, the answer is (-7,9), which is highly suspect IMO-regardless of order I should NOT get any negative numbers via multiplication, as the matrix for reflection has all positive numbers- when I worked out reflection followed by translation, I got (15,11).

So, for a transformation AB, do you apply B first then A? And in the second case, did I get mixed up again or did the answer key get it wrong?

Answer 1

Yes, the order of $AB$ is usually $B$ then $A$. It's like a composite function $f\circ g(x)$, you apply $g$ first, then $f$. The answer of that next problem should be $(14,11)$.

Answer 2

What you are dealing are affine transformations in the plane, i.e. transformations of the form: $$ \vec{y}=A \vec{x}+\vec{t} $$ where $A$ is a $2 \times2$ real matrix: $$ A= \left[ \begin {array}{ccccc} a & b\\ c & d \end {array} \right] $$ and $\vec{t}$ is a vector representing the translation. In your case: $$ \vec{t}= \left[ \begin {array}{ccccc} t_1\\ t_2 \end {array} \right] = \left[ \begin {array}{ccccc} 5\\ 4 \end {array} \right] $$ As you can see here such transformations can be represented by $3 \times 3$ matrices of the form: $$ M= \left[ \begin {array}{ccccc} a&b & t_1\\ c & d & t_2\\ 0 & 0& 1 \end {array} \right] $$ that operates as the usual rows-columns product rule, giving: $$ \left[ \begin {array}{ccccc} y_1\\ y_2\\ 1 \end {array} \right]= \left[ \begin {array}{ccccc} a&b & t_1\\ c & d & t_2\\ 0 & 0& 1 \end {array} \right] \left[ \begin {array}{ccccc} x_1\\ x_2\\ 1 \end {array} \right] = \left[ \begin {array}{ccccc} ax_1+bx_2+t_1\\ cx_1+dx_2+t_2\\ 1 \end {array} \right] $$ Where we have add a third component $1$ to the vectors $\vec x$ and $\vec y$.

In your case you have: $$ T= \left[ \begin {array}{ccccc} 0&0 &5\\ 0& 0 & 4\\ 0 & 0& 1 \end {array} \right] \qquad R= \left[ \begin {array}{ccccc} 0&-1 &0\\ 1& 0 & 0\\ 0 & 0& 1 \end {array} \right] \qquad M= \left[ \begin {array}{ccccc} 0&1 &0\\ 1& 0 & 0\\ 0 & 0& 1 \end {array} \right] $$ For combining two transformation you have to calculate the product of their matrices and we know that the product of matrices is not commutative, so that e.g. $TR \ne RT$ ( as you can be easily test).

Multiplying the given matrices with the given vector $$ \mathbf{v}= \left[ \begin {array}{ccccc} 7\\ 9\\ 1 \end {array} \right] $$ in the right order you can easily find the answers to all your questions.