Describe displacement with fixed-axis components, combine compatible vectors, and use small exact examples to introduce span, independence, basis and real vector spaces.
Key knowledge
A planar Euclidean vector can be represented by ordered components in a chosen basis. Vector addition adds corresponding components, and scalar multiplication multiplies every component by the scalar.
Values and equations
u + v = (uₓ + vₓ, uᵧ + vᵧ); λv = (λvₓ, λvᵧ)
u, v
Planar vectors in a common basis
λ
Real scalar
|v| = √(vₓ² + vᵧ²)
vₓ, vᵧ
Components in an orthonormal Cartesian basis
|v|
Euclidean magnitude
Conditions and scope
Use the same basis for component operations.
In an orthonormal Cartesian basis the magnitude is √(vₓ² + vᵧ²); this formula does not apply unchanged to a nonorthonormal basis.
Physical vectors being added must have compatible units.
Sources · 8
References
Sources retain their recorded scope. A source link alone does not establish verification of every statement.
Two-dimensional Cartesian vector operations and norm.
2026-10-10 · Checked component formulas; scope excludes general abstract vector spaces and curved-space transport.
Foundational review
2026-10-10 · Fresh primary-source inspection of the complete compact record, including every statement, equation, quantity and material condition, followed by independent AI editorial review. This check does not certify all linked lessons or every passage of every cited publication.
Sources from other places
Scope or convention differs
Cartesian components add and scale componentwise, with the Pythagorean norm.
The source uses standard axes; repository basis and compatible-unit conditions prevent overgeneralising its norm formula.
University of Cape Town, Department of Mathematics and Applied Mathematics
A planar Euclidean vector can be represented by ordered components in a chosen basis. Vector addition adds corresponding components, and scalar multiplication multiplies every component by the scalar.
A real linear map T:ℝⁿ→ℝᵐ satisfies T(αu+βv)=αT(u)+βT(v). After choosing input and output bases, it is represented by an m×n matrix A: the output coordinates are y=Ax, with yᵢ=Σⱼ₌₁ⁿaᵢⱼxⱼ.
For x,y∈Rⁿ, ⟨x,y⟩=Σxᵢyᵢ and ‖x‖=√⟨x,x⟩. Vectors are orthogonal when their inner product is zero; the product is symmetric, bilinear and positive definite.
A regular C¹ plane or space curve r(t), a≤t≤b, has length ∫ₐᵇ‖r′(t)‖dt. For a C² regular space curve its curvature is κ=‖r′×r″‖/‖r′‖³, equivalently ‖dT/ds‖ with unit tangent T and arc length s.
Displacement is the final position minus the initial position in a stated frame. Distance is the length of the travelled path and is at least the displacement magnitude. Average velocity is displacement divided by elapsed time; average speed is distance divided by that time. A round trip can have zero average velocity and positive average speed.
This relation concerns the stated displacement, distance and average speed claim and its conditions; it does not assert an unqualified prerequisite or an additional theorem.
Learning guidance · 2 / 2
Preserved learning suggestions. They are not proof, empirical evidence or mandatory prerequisites.