Answer:
[tex]\begin{bmatrix}-7 \\-2\end{bmatrix}[/tex] (might also be denoted as [tex]\langle-7,\, -2\rangle[/tex] in some textbooks.)
Step-by-step explanation:
Let [tex]a[/tex] denote the position vector of point [tex]A[/tex] [tex](2,\, 5)[/tex]. Thus:
[tex]a = \begin{bmatrix}2 \\ 5\end{bmatrix}[/tex].
Similarly, let [tex]a^{\prime}[/tex] denote the position vector of point [tex]A^{\prime}[/tex]. Thus:
[tex]\displaystyle a^{\prime} = \begin{bmatrix}-5 \\ 3\end{bmatrix}[/tex].
Let [tex]x[/tex] be the translation vector from point [tex]A[/tex] to point [tex]A^{\prime}[/tex]. Adding this translation vector [tex]x\![/tex] to the position vector of the original point should yield the positional vector of the new point. In other words:
[tex]a + x = a^{\prime}[/tex].
[tex]\begin{aligned}x &= a^{\prime} - a \\ &= \begin{bmatrix}-5 \\ 3\end{bmatrix} - \begin{bmatrix}2 \\ 5\end{bmatrix}\\ &= \begin{bmatrix}-5 - 2 \\ 3 - 5\end{bmatrix} \\ &=\begin{bmatrix}-7 \\ -2\end{bmatrix}\end{aligned}[/tex].
Thus, the required translation vector would be [tex]\begin{bmatrix}-7 \\ -2\end{bmatrix}[/tex] (or equivalently, [tex]\langle -7,\, -2 \rangle[/tex].
