Vectors and translations in two dimensions
Geometry and measures • Vectors
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Key concepts
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Definition of a vector
A vector is a quantity with both magnitude and direction. A vector does not depend on its position; only the length and direction matter. A vector is represented by an arrow in a diagram or by an ordered pair of numbers in column or bracket form. The components show how far to move horizontally and vertically.
Notation and components
A 2D vector has two components, written as (x, y) or as a column [x; y]. The first component indicates horizontal movement: positive to the right and negative to the left. The second component indicates vertical movement: positive upwards and negative downwards. Column notation supports calculation: adding a column vector [x; y] to point coordinates (a, b) produces (a + x, b + y).
Diagrammatic representation
Diagrammatic vectors appear as directed line segments (arrows). Arrow length shows magnitude; arrow direction shows orientation. An arrow placed at any starting point represents the same vector if length and direction match. Diagrammatic drawing clarifies translations: drawing the arrow from a point shows the image point at the arrow tip after translation. Parallel arrows of equal length show identical vectors acting on different points.
Column representation and operations
Column form lists components vertically, for example [3; -2]. Addition of vectors occurs component-wise: [a; b] + [c; d] = [a + c; b + d]. Scalar multiplication scales components: k[a; b] = [ka; kb]. Column form links directly to coordinates: translating point (x, y) by [a; b] produces (x + a, y + b). Calculation becomes arithmetic on coordinates.
Translations as 2D vectors
A translation shifts every point of a shape by the same 2D vector. The translation [p; q] moves each point (x, y) to (x + p, y + q). The shape’s size and orientation remain unchanged. Translations are rigid motions, so lengths and angles remain constant. Only position changes; reflections, rotations and scalings do not occur during a translation.
Key notes
Important points to keep in mind