Transformations: Question 3

Worked solution

Understanding a translation by a column vector

A translation by the column vector $\begin{pmatrix} a \\ b \end{pmatrix}$ moves every point $a$ units in the $x$-direction and $b$ units in the $y$-direction.

In symbols, the image of a point $(x,\ y)$ is:

$\left(x,\ y\right) \longmapsto \left(x + a,\ y + b\right)$

Apply it to this point

Here the object point is $P = (-3,\ 4)$ and the translation vector is $\begin{pmatrix} 5 \\ -6 \end{pmatrix}$, so $a = 5$ and $b = -6$.

New $x$-coordinate (add the top entry):

$x' = -3 + 5 = 2$

New $y$-coordinate (add the bottom entry, which is negative):

$y' = 4 + (-6) = 4 - 6 = -2$

Therefore:

$P' = (2,\ -2)$

Check with a sketch

Starting at $(-3,\ 4)$, move right $5$ (because the top number is $+5$): the $x$-coordinate goes from $-3$ to $2$. Then move down $6$ (because the bottom number is $-6$): the $y$-coordinate goes from $4$ to $-2$. This lands exactly on $(2,\ -2)$, confirming the result.

Final answer

$\boxed{P' = (2,\ -2)} \quad\Rightarrow\quad \textbf{B}$