Cross Product
Definition and meaning of Cross Product in chemistry.
Cross product is a vector operation, written A × B, that combines two vectors to produce a third vector perpendicular to both, with magnitude |A||B|sinθ, where θ is the angle between them.
In more detail
The direction of the resulting vector is given by the right-hand rule: curling the fingers from A toward B, the thumb points along A × B. Unlike the dot product, which yields a scalar related to cosθ, the cross product yields a vector and vanishes when A and B are parallel (θ = 0°). In chemistry, cross products appear wherever rotational or directional quantities are combined, such as angular momentum, torque, and the quantum mechanical angular momentum operator used in atomic and molecular orbital theory.
Key facts
| Formula | A × B = |A||B| sinθ n̂ |
|---|---|
| Also known as | Vector product |
| Key applications | Angular momentum (L = r × p), torque, magnetic dipole interactions |
| Field | Physical Chemistry |
The orbital angular momentum of an electron is defined as L = r × p, the cross product of its position vector r and linear momentum vector p; this quantity underlies the quantization of angular momentum in the Bohr and quantum mechanical models of the atom.
Frequently asked questions
How is the cross product different from the dot product?
The cross product of two vectors gives a vector perpendicular to both (magnitude proportional to sinθ), while the dot product gives a scalar (proportional to cosθ) representing how much the vectors align.
Why does the cross product matter in chemistry?
It is essential for describing rotational and orientation-dependent quantities, such as orbital angular momentum, torque on a dipole in a field, and the commutation relations of angular momentum operators in quantum chemistry.