unit vector — from formula anatomyA unit vector is a vector with magnitude exactly equal to one, used to indicate direction without any contribution to magnitude. It is typically denoted with a hat (e.g., r̂, θ̂, φ̂) and is defined as any vector divided by its own length: v̂ = v/|v|. Unit vectors form the basis for constructing coordinate systems, such as Cartesian (î, ĵ, k̂), spherical (r̂, θ̂, φ̂), and cylindrical (ρ̂, φ̂, ẑ), and they allow physical quantities like velocity and force to be decomposed into directional components.
The concept of a unit vector emerged with the development of vector analysis in the 19th century, particularly through the work of Josiah Willard Gibbs and Oliver Heaviside, who formalized vector notation for physics.
In practice, a unit vector is obtained by normalizing a vector: divide each component by the vector's Euclidean norm. In curvilinear coordinates, unit vectors are position-dependent; for example, in spherical coordinates, r̂ points radially outward, while θ̂ and φ̂ change direction as you move. This property is crucial for correctly expressing derivatives like the gradient and divergence in non-Cartesian systems.
Unit vectors are used everywhere in physics and engineering: in mechanics to define directions of forces and velocities, in electromagnetism to express fields, and in computer graphics for surface normals and lighting calculations.
In quantum mechanics, the unit vector in Hilbert space represents a pure state, and its normalization condition (⟨ψ|ψ⟩ = 1) is a direct analog of the classical unit vector's magnitude being one.
| \vec{v}_{\text{sky}} | variable | tangential (sky-projected) velocity — velocity metre per second |
| \vec{v} | variable | total velocity vector — velocity metre per second |
| \hat{d} | variable | unit vector along the line of sight — unit vector |
| \cdot | operator | scalar product of two vectors — dot product |
| A | variable | Polarization vector — polarization vector |
| Δ | variable | Amplitude of the field — amplitude volt per metre |
| e_x | variable | Unit vector along x-axis — unit vector |
| e_y | variable | Unit vector along y-axis — unit vector |
| i | operator | Complex number with square -1 — imaginary unit |
| m_1 | variable | mass of the first compact object — mass kilogram |
| m_2 | variable | mass of the second compact object — mass kilogram |
| \boldsymbol{\chi}_1 | variable | dimensionless spin vector of the first object — dimensionless spin |
| \boldsymbol{\chi}_2 | variable | dimensionless spin vector of the second object — dimensionless spin |
| \boldsymbol{L} | variable | unit vector along orbital angular momentum — unit vector |
| \cdot | operator | computes the scalar projection of one vector onto another — dot product |
| \mathbf{F} | variable | gravitational force vector — force newton |
| m_1 | variable | mass of first object — mass kilogram |
| m_2 | variable | mass of second object — mass kilogram |
| r | variable | distance between the centers of mass — distance metre |
| \hat{\mathbf{r}} | variable | unit vector pointing from one mass to the other — unit vector |
| G | constant | Newtonian gravitational constant, fundamental constant of gravitation — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared |
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