Position vector

As explained in Geometry for Programmers

A point vector is a vector conventionally interpreted as representing a point by its components from the origin.

A position vector represents a point by conventionally starting the vector at the origin. The vector keeps its direction and length, while its coordinates identify the point it reaches. This convention connects vector arithmetic with geometric operations. section 9.1 section 2.1.2

Use a position vector to:

• Write a point as a coordinate tuple, such as (0.5, 1.5) in 2D or (0.5, 1.5, 0.0) in 3D. section 2.1.2
• Express the same vector as scaled basis vectors, such as (0.5, 1.5) = 0.5i + 1.5j. section 2.1.2
• Translate a model by adding one vector to all its point vectors, or scale it by multiplying them by a scalar. section 9.1

A point vector in practice

Treating a vector toward (0.5, 1.5) as a position vector lets that coordinate describe a point while retaining vector operations such as addition and scalar multiplication. section 2.1.2

Figure 9.1 A model made of point vectors is translated by vector addition.

Shows how adding the same vector to every point vector translates a geometric model. section 9.1

How Geometry for Programmers covers it

How deep it goes

Defines the origin convention, coordinate forms, basis decomposition, translation, and scaling, but does not develop vector products.

Examples

Uses (0.5, 1.5), its 3D extension, basis form 0.5i + 1.5j, and point-vector models undergoing translation or scaling.

What it assumes

Builds the distinction between points and vectors from basic definitions, assuming only coordinate tuples and real-number operations.

What you can do after

Evaluate when a vector can conventionally represent a point and apply vector addition or scalar multiplication to point-vector models.

How it connects

Links points with vectors, coordinate tuples, basis vectors, vector addition, scalar multiplication, translation, and scaling.

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Also explained in

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  Grokking AI Algorithms, Second EditionA position is the point or vector in a search space that represents a particle's candidate solution in particle swarm optimization.
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  Optimization Algorithms: AI techniques for design, planning, and control problemsA permutation encoding the candidate solution represented by a particle in permutation-based particle swarm optimization.

Compare how 3 books explain position vector →

Related concepts

• vectorkind of · A vector is an ordered collection of components representing an element of a vector space.
• particlepart of · A particle is an individual entity represented or managed as a unit within a physical or computational model.
• vectorkind of · A vector is an ordered, indexed sequence of elements, often numerical, represented as a mathematical quantity or a computational collection.
• rayused by · A ray is a geometric object with one endpoint that extends infinitely in a single direction.
• particle swarm optimizationpart of · Particle swarm optimization is a population-based optimization method in which candidate solutions adjust their positions using individual and collective search information.
• permutation problemequivalent to · A problem whose candidate solutions are ordered arrangements of a specified set of elements.

This explainer was generated by Manning's knowledge graph from the text of Geometry for Programmers. The section links point to the parts of the book it draws on.