Translation by a vector: the transformation that 'slides' everything

Definition

The translation by vector $\vec{u}$ maps each point $M$ to $M'$ such that:

$\vec{MM'} = \vec{u}$

Every point moves by the same vector.

Image of point M

To build $M'$:

1. Draw a representative of $\vec{u}$ starting at $M$.
2. The arrow tip is $M'$.

In coordinates: if $\vec{u}(a, b)$ and $M(x, y)$, then $M'(x + a, y + b)$.

No fixed point

Unlike rotation or reflection, translation has no invariant point, except when $\vec{u} = \vec{0}$ (then it’s the identity).

Properties

Translation preserves:

• Lengths
• Angles
• Areas
• Orientation
• Parallelism (a line remains parallel to its image)

Translation maps:

• A line to a parallel line
• A segment to a same-length segment
• A circle to a same-radius circle

Composition

$t_{\vec{u}} \circ t_{\vec{v}} = t_{\vec{u} + \vec{v}}$

Two successive translations = one translation by the sum vector.

Common mistakes

1. Confusing with rotation: translation has no center.
2. Believing there’s a fixed point: no, except for zero translation.
3. Forgetting vector direction: $\vec{u}$ and $-\vec{u}$ give opposite translations.

?À toi de jouer

Point A(2, 3) translated by u(4, −1). Coordinates of A'?

A.(6, 2)B.(6, 4)C.(2, 4)D.(−2, 4)

Valider