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? What translation vector represents the transformation from A (2,5) to A'(-5,3) ? 2​

? What translation vector represents the transformation from A (2,5) to A'(-5,3) ? 2​-example-1
User Pjmorse
by
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1 Answer

9 votes

Answer:


\begin{bmatrix}-7 \\-2\end{bmatrix} (might also be denoted as
\langle-7,\, -2\rangle in some textbooks.)

Explanation:

Let
a denote the position vector of point
A
(2,\, 5). Thus:


a = \begin{bmatrix}2 \\ 5\end{bmatrix}.

Similarly, let
a^(\prime) denote the position vector of point
A^(\prime). Thus:


\displaystyle a^(\prime) = \begin{bmatrix}-5 \\ 3\end{bmatrix}.

Let
x be the translation vector from point
A to point
A^(\prime). Adding this translation vector
x\! to the position vector of the original point should yield the positional vector of the new point. In other words:


a + x = a^(\prime).


\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}.

Thus, the required translation vector would be
\begin{bmatrix}-7 \\ -2\end{bmatrix} (or equivalently,
\langle -7,\, -2 \rangle.

User Gnicholas
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