Plane vectors in a fixed basis

Mathematical statement

In the fixed standard basis, a plane vector is u=(u₁,u₂)∈R². Define u+v=(u₁+v₁,u₂+v₂), k u=(k u₁,k u₂), and u−v=(u₁−v₁,u₂−v₂). Two vectors are equal exactly when both corresponding components are equal; the zero vector is (0,0).

Representative formula (exact TeX)

u+v

Assumptions

• The basis is fixed, and components and scalars are real. The exercises use exact integer-component and integer-scalar instances.
• Ordered pairs here represent vector components; geometric points, position vectors, and arbitrary coordinate systems are not identified as one object.

Symbol scope

• u — first plane vector
• v — second plane vector

Intuition, definition, and misconceptions

Add or subtract corresponding components; comparing only their sums is insufficient. The order in u−v matters, and exchanging the operands gives the opposite vector.

The zero scalar gives the zero vector, while a negative scalar gives the algebraic representation of direction reversal. Addition and scalar multiplication are distinct from an inner product, and their results remain vectors in R².

This pulse does not infer lengths, angles, unit vectors, or basis changes from component operations. General vector spaces, their axioms, and physical units have separate obligations.

Common errors include exchanging vector coordinates, treating equal lengths as vector equality, or confusing scalar multiplication with an inner product.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Real number — Vector components and scalars belong to the reals.
• Arithmetic operations — Componentwise vector operations use addition, subtraction, and multiplication.
• Coordinate plane — Fix the ordered-component and basis setting.

Verification boundary

Finite integer-component operations can be recomputed exactly; general vector-space axioms and the geometric semantic bridge have no attached Evidence.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bind the componentwise structure of Fin 2→Real or Real×Real and its fixed-basis representation.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.