Vector Cross Product Identities

Vector calculus identities regarding operations on vector fields such as divergence gradient curl etc.

Vector cross product identities. The dot product of two vectors is a scalar but the dot product of a pseudovector and a vector is a pseudoscalar so the scalar triple product must be pseudoscalar valued. This also relates to the handedness of the cross product. The pythagorean trigonometric identity then provides. Vector algebra relations regarding operations on individual vectors such as dot product cross product etc.

6 components 6 votes and their total is the cross product. The cross product is the area of the parallelogram spanned by the vectors aand b. The abbreviations used are. .

When the vectors are crossed each pair of orthogonal components like ax by a x b y casts a vote for where the orthogonal vector should point. There are two lists of mathematical identities related to vectors. Unit vectors enable two convenient identities. The cross product is linear in each factor so we have for example for vectors x y u v ax by cu dv acxu adxv bcy u bdy v.

The generalization of the dot product formula to riemannian manifolds is a defining property of a riemannian connection which differentiates a vector field to give a vector valued 1 form. The dot product of two unit vectors yields the cosine which may be positive or negative of the angle between the two unit vectors. The cross product satisfies the general identity 7 note that is not a usual polar vector but has slightly different transformation properties and is therefore a so called pseudovector arfken 1985 pp. Here the notation a b denotes the vector cross product of vectors a and b.

Cross product rule. The magnitude of the cross product of the two unit vectors yields the sine which will always be positive. The cross product transforms as a pseudovector under parity transformations and so is properly described as a pseudovector.