Describing translations as 2D vectors
A vector in 2D is a quantity with magnitude AND direction. The translation of a shape can be described by a 2D vector specifying horizontal and vertical movement.
Vector notation
Common forms:
- Column vector: (a / b) — top is horizontal (right is +); bottom is vertical (up is +).
- Notation: a = (3 / −2) means "right 3, down 2".
- Bold or underlined letters: a, b, c.
- Lowercase with arrow: ⃗AB.
Translating a shape
Every point of the shape moves by the same vector.
Worked example: triangle vertices (1, 1), (3, 1), (2, 4) translated by (2 / −1).
- (1+2, 1−1) = (3, 0).
- (3+2, 1−1) = (5, 0).
- (2+2, 4−1) = (4, 3).
Magnitude and direction
The magnitude (length) of vector (a / b) = √(a² + b²) (Pythagoras).
Direction can be given as a bearing or as an angle from the horizontal.
Negative and zero vectors
- Zero vector: (0 / 0) — no movement.
- Negative vector: −a is the same magnitude but opposite direction.
Combining vectors (G25)
Vector addition (head-to-tail) and scalar multiplication appear in G25.
✦Worked example— Worked examples
Example 1. Describe the translation that maps (3, 5) to (7, 2).
- Vector = (7−3, 2−5) = (4 / −3).
Example 2. Apply translation (−2 / 5) to (8, 1).
- Image = (8−2, 1+5) = (6, 6).
⚠Common mistakes
- Confusing column vector with coordinate — coordinates use (x, y); vectors use (top / bottom).
- Sign errors in vertical — up is positive; down is negative.
- Forgetting magnitude needs Pythagoras — magnitude is a length, not just |a| + |b|.
- Wrong direction of subtraction — to find translation A → B, do B − A.
- Mixing translation with rotation/reflection — translation is purely a slide, no rotation.
➜Try this— Quick check
A translation maps (2, 7) to (−1, 4). State the column vector.
- Vector = (−1 − 2, 4 − 7) = (−3 / −3).
AI-generated · claude-opus-4-7 · v3-deep-geometry