Translations as Vectors

Translations as Vectors

A vector describes both a magnitude (size) and a direction. A translation moves every point of a shape by the same amount in the same direction — this displacement is represented exactly by a vector.

Column Vector Notation

A translation is written as a column vector \( \begin{pmatrix} x \ y \end{pmatrix} \) where \( x \) is the horizontal displacement (positive = right, negative = left) and \( y \) is the vertical displacement (positive = up, negative = down).

\[ \text{Translation by } \begin{pmatrix} 3 \ -2 \end{pmatrix} \text{ moves every point 3 right and 2 down.} \] x y A B C A' B' C' \( \begin{pmatrix}4\-3\end{pmatrix} \)

Properties of Translations

  • Every point moves by the same vector — the shape and size are unchanged (isometric).
  • The image is congruent to the original and the same orientation.
  • Vectors are denoted in bold \( \mathbf{a} \) or with an arrow \( \vec{AB} \). In hand-written work, underline: \( \underline{a} \).
  • The reverse translation is the negative vector: \( -\begin{pmatrix}x\y\end{pmatrix}=\begin{pmatrix}-x\-y\end{pmatrix} \).

 Key Takeaways

  • Column vector \( \begin{pmatrix}x\y\end{pmatrix} \): \( x \) = horizontal (+ right), \( y \) = vertical (+ up).
  • Translation is isometric — congruent image, same orientation.
  • Reverse translation: negate both components.
  • Vectors are equal if they have the same magnitude and direction — position on the grid does not matter.