Matrix translation by (1x2) vector

Question

I'm having trouble figuring out how to approach this matrix translation question:

Find the equation of the image line produced by translating all of the points on the line $y = 3x -1$ by the vector $\begin{smallmatrix} 2 \\ 3 \end{smallmatrix}$.

I thought this worked out to be:

$$ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\begin{pmatrix} k \\ 3k -1 \end{pmatrix} $$

But I'm obviously mistaken because you can't multiply a (1x2) matrix by another (1x2) matrix. What am I missing?

score 1 · Accepted Answer · answered Jul 12 '14 at 12:08

1

I am quite sure that what is meant here is the following:

$$\begin{pmatrix} x'\\y'\\\end{pmatrix}=\begin{pmatrix}x\\y\\\end{pmatrix}+\begin{pmatrix}2\\3\\\end{pmatrix} $$ Where $y=3x-1$, so that we have $$\begin{pmatrix} x'\\y'\\\end{pmatrix}=\begin{pmatrix}x\\3x-1\\\end{pmatrix}+\begin{pmatrix}2\\3\\\end{pmatrix}= \begin{pmatrix}x+2\\3x+2\\\end{pmatrix}$$

The point is that 'translating by a vector' just means 'adding a vector', when talking bout some general vector $\begin{pmatrix}x\\y\\\end{pmatrix}$.

answered Jul 12 '14 at 12:08

Danu

1,772

The answer given is $y = 3x - 4$ – hohner Jul 12 '14 at 12:10
@hohner This is the same answer that I obtain. I will let you discover it for yourself: How can one produce $y'$ from $x'$? – Danu Jul 12 '14 at 12:12
Think I've got it: $y' = 3(x + 2) - 4$ – hohner Jul 12 '14 at 12:17

Matrix translation by (1x2) vector

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