Translations calculated using matrices

Question

You have used matrices to calculate the results of certain rotations and reflections. Which ones? Are translations calculated using matrices?

I know of the different types of reflections, like across the x-axis, y-axis, origin, and y=x, but I haven’t heard of using matrices to calculate translations… does someone mind explaining to me? Thank you in advance!

score 0 · Answer 1 · answered Aug 17 '18 at 05:51

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No, translations can't be computed via the same matrix-multiplication process.

The reason is that matrix multiplication will always map the vector $\vec0$ to itself, whereas a translation doesn't (except when the translation vector is $\vec0$ itself).

For all $a,b,c,d,\qquad$ $$\pmatrix{a & b \\c & d\\}\pmatrix{0 \\0 \\}=\pmatrix{0 \\0 \\}$$ But $$\pmatrix{0 \\0 \\}+\vec v=\vec v$$

answered Aug 17 '18 at 05:51

Arnaud Mortier

27,276

Thank you! This helped a lot, but why does it result in the vector when the matrix is added? @arnaud mortier – Patricia M. Aug 17 '18 at 06:02
@PatriciaM. In the $2\times 2$ matrix case above, the matrix isn't added, it's multiplied. Just like in the case of rotations and reflections. See also this perhaps more readable for a first time. – Arnaud Mortier Aug 17 '18 at 06:55

score 0 · Answer 2 · answered Aug 17 '18 at 06:18

If you want to describe a translation in say $\mathbb{R}^2$ you may use a $3\times3$ matrix as follows:

For a translation by a vector $\vec{x}=(a,b)$ , $T_{\vec{x}}=\begin{pmatrix} 1 & 0 & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}$

So $T_{\vec{x}}\cdot (\alpha,\beta,1)^t=(a+\alpha,b+\beta,1)$ which is like $(\alpha,\beta)+\vec{x}=(a+\alpha,b+\beta)$

Translations calculated using matrices

2 Answers2