dot product — from formula anatomyThe dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers (usually vectors) and returns a single number. It is defined as the sum of the products of the corresponding entries. Geometrically, it equals the product of the magnitudes of the two vectors and the cosine of the angle between them. This makes it a fundamental tool for measuring the alignment of vectors.
The dot product was formalized in the 19th century by mathematicians like Hermann Grassmann and Josiah Willard Gibbs, who developed vector calculus.
Given two vectors a = (a1, a2, ..., an) and b = (b1, b2, ..., bn), the dot product is a1*b1 + a2*b2 + ... + an*bn. If the vectors are perpendicular, the dot product is zero; if they point in the same direction, it is positive and equals the product of their lengths. This property is used to project one vector onto another and to determine angles in multi-dimensional spaces.
It is used everywhere in physics and engineering, from computing work done by a force (force dot displacement) to machine learning algorithms for similarity measures.
In quantum mechanics, the dot product of state vectors gives the probability amplitude for transitions between states.
| \rho_{\rm GJ} | variable | Goldreich-Julian charge density — charge density coulomb per cubic metre |
| \mathbf{\Omega} | variable | Angular velocity vector of the neutron star — angular velocity radian per second |
| \mathbf{B} | variable | Magnetic field vector — magnetic field tesla |
| c | constant | Universal constant, speed of light in vacuum — speed of light = 299792458 metre per second |
| 2\pi | constant | Mathematical constant, 2π — two pi = 6.283 |
| \cdot | operator | Scalar product of two vectors — dot product |
| \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 |
| \tilde{\mu} | variable | Corrected distance modulus or magnitude — distance modulus |
| \mu | variable | Original distance modulus or magnitude — distance modulus |
| A_D | variable | Amplitude of dipole anisotropy — dimensionless |
| \hat{n} | variable | Unit vector in the direction of observation — direction |
| \hat{p} | variable | Unit vector pointing toward the dipole axis — direction |
| \cdot | operator | Scalar product of two vectors — dot product |
| E | variable | magnetic energy of the moment — energy joule |
| \boldsymbol{\mu} | variable | magnetic dipole moment — magnetic dipole moment ampere square metre |
| \mathbf{B} | variable | magnetic field — magnetic field tesla |
| \cdot | operator | takes the scalar product of two vectors — dot product |
| 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 |
| H_{\text{HFI}} | variable | magnetic hyperfine interaction Hamiltonian — hamiltonian joule |
| \boldsymbol{\mu}_I | variable | nuclear magnetic moment — magnetic moment ampere square metre |
| \mathbf{A} | variable | hyperfine coupling tensor — tensor joule per ampere square metre |
| \mathbf{S} | variable | electron spin operator — spin operator |
| \cdot | operator | scalar product of vectors — dot product |
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