cross product

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cross product

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cross product

Definition/Summary

The cross product of two vectors

Click to see the LaTeX code for this image

and

is a third vector (strictly, a pseudovector or axial vector)

perpendicular to both of the original vectors, with magnitude equal to the product of their magnitudes times the (positive) sine of the angle between them, and in the direction determined by the right-hand rule.

It is anti-commutative:

If the vectors

and

are considered as 1-forms, then the wedge product

is a 2-form (a directed area), and its dual

is a dual 1-form, corresponding to the pseudovector

Equations

The magnitude of

is the area of the paralleogram with sides

and

In terms of Cartesian components:

The triple product (scalar product)

is the volume of the parallepiped with sides

and

, and therefore:

The repeated cross product

is a vector perpendicular to

, and is a linear combination of

and

Scientists

Recent forum threads on cross product

Breakdown

Mathematics
> Algebra
>> Vector

Images

Extended explanation

For two vectors

, the cross poduct can be written as the determinant of a 3x3 matrix:

Where

is a right-handed orthonormal basis.

Polar vectors and pseudovectors:

A polar (ordinary) vector is reversed under an inversion of the coordinate axes:

However, the cross product of two polar vectors is not reversed:

In other words, the cross product of two polar (ordinary) vectors is invariant (the same) under an inversion of the coordinate axes: this is called a pseudovector (axial vector).

Directed area:

A directed area is a flat surface together with a magnitude (its area), and a sign (

) indicating a direction of rotation within the surface (alternatively, indicating a preferred normal direction).

A directed area is an elementary 2-form (a wedge product of two 1-forms).

A general 2-form is a sum of directed areas.

Its dual (in three-dimensional space) is a 1-form in the dual space, corresponding to a pseudovector normal to the surface and with magnitude equal to its area.

Triple product 3-form pseudoscalars and directed volume:

The wedge product of three 1-forms is a 3-form.

In three-dimensional space, all 3-forms are multiples of each other, and so a 3-form is essentially a scalar. To be precise, a 3-form is the dual of a 0-form, which is a scalar.

However, under an inversion of the coordinate axes, a 3-form is multiplied by minus-one, and so a 3-form technically is a psuedoscalar.

The triple product

of three vectors is the dot product of a vector and a pseudovector, and is the pseudoscalar equal to the wedge product

of the three vectors, in the same order. It is the directed volume of the parallepiped whose sides are those three vectors ("directed" because it includes a sign (

) indicating a direction of rotation around the common vertex).

Commentary

tiny-tim @ 06:39 AM Feb15-09

Added wedge product directed area directed volume and pseudoscalars, and rewrote pseudovectors in ext expl.