Cross Product

The cross product is a binary operation on vectors in an oriented three-dimensional Euclidean space. Written $\mathbf a\times\mathbf b$, it produces a vector perpendicular to both inputs, with magnitude equal to the area of the parallelogram they span and direction determined by the right-hand rule. Unlike the dot product, which produces a scalar, the cross product produces a vector. It is widely used in geometry and physics to describe oriented areas, perpendicular directions, and rotational quantities. (ocw.mit.edu)

Geometric definition

For nonzero vectors $\mathbf a$ and $\mathbf b$ making an angle $\theta\in[0,\pi]$, the cross product has magnitude

When the vectors are not parallel, its direction is fixed by the right-hand rule: curling the fingers of the right hand from $\mathbf a$ toward $\mathbf b$ through the smaller angle makes the thumb point along $\mathbf a\times\mathbf b$. Equivalently, the ordered triple $(\mathbf a,\mathbf b,\mathbf a\times\mathbf b)$ is positively oriented. The magnitude is the parallelogram’s base times its height. (web.mit.edu)

If either input is zero, or the inputs are parallel or antiparallel, their cross product is zero. Thus

This includes the zero-vector cases; a zero cross product has no direction. For fixed input magnitudes, perpendicular inputs give the greatest possible cross-product magnitude. (openstax.org)

Coordinate formula

In a positively oriented orthonormal basis $(\mathbf e_1,\mathbf e_2,\mathbf e_3)$, let

Then

The basis vectors obey

Reversing either pair changes the sign, while crossing a basis vector with itself gives zero. (ocw.mit.edu)

A common computational mnemonic is the formal determinant

Here the first row contains vectors rather than scalar entries: the notation means to expand along that row, using the alternating cofactor signs. It is not an ordinary scalar-valued determinant. (ocw.mit.edu)

For example, direct substitution gives

The result is perpendicular to both inputs, as verified by

Algebraic properties

The cross product is bilinear: it is linear in each input separately. For real scalars $\alpha,\beta$,

with an analogous identity in the second input. It is also alternating and anticommutative:

Consequently, multiplication order matters. (web.mit.edu)

The cross product is not associative. Its nested products instead satisfy the vector triple-product identities

Parentheses therefore cannot generally be omitted from repeated cross products. Its magnitude and the dot product are related by Lagrange’s identity:

This identity expresses the squared area spanned by two vectors entirely in terms of their lengths and dot product. (openstax.org)

Triple products and geometric applications

The scalar triple product combines a dot product with a cross product:

It is the signed volume of the parallelepiped generated by the three vectors; its absolute value is the ordinary volume. Cyclically permuting the inputs leaves it unchanged, while exchanging two inputs reverses its sign. A zero scalar triple product means that the three vectors are linearly dependent. (ocw.mit.edu)

For noncollinear points $A,B,C$, set

Then $\mathbf u\times\mathbf v$ is normal to their plane, and the triangle’s area is

A unit normal is

Changing the order of the points reverses the normal without changing the area. This construction fails for collinear points because its denominator is zero. (openstax.org)

Orientation and transformation behavior

The cross product requires both a Euclidean inner product and a choice of orientation. Its usual coordinate formula assumes a right-handed orthonormal basis; reversing the chosen orientation reverses the product’s sign. (cs.cmu.edu)

For an orthogonal transformation $Q$, its transformation law is

Thus it transforms like an ordinary vector under rotations, for which $\det(Q)=1$, but acquires an extra sign under orientation-reversing transformations. When the inputs are ordinary polar vectors, their cross product is consequently called an axial vector or pseudovector. In particular, spatial inversion changes both inputs’ signs but leaves their cross product unchanged:

This distinction concerns transformation behavior, not whether the result belongs to a vector space. (sites.science.oregonstate.edu)

Uses in mechanics

In classical mechanics, cross products encode the direction and magnitude of rotational quantities. The torque exerted by a force $\mathbf F$ about a chosen origin is

where $\mathbf r$ runs from that origin to the force’s application point. A particle’s angular momentum about the same origin is

where $\mathbf p$ is its linear momentum. Both depend on the reference point. Their axial directions represent the orientation of rotational effects rather than a direction of translational motion. (feynmanlectures.caltech.edu)

For a rigid body rotating with angular velocity $\boldsymbol\omega$, the velocity relative to a point on its fixed rotation axis is

The sine factor in the cross-product magnitude automatically selects the perpendicular distance to the axis. (feynmanlectures.caltech.edu)

Higher-dimensional generalizations

The exterior product provides a related operation in any dimension. For vectors $\mathbf a,\mathbf b$, the wedge product $\mathbf a\wedge\mathbf b$ is a bivector representing an oriented area, rather than generally another vector. In oriented three-dimensional Euclidean space, the Hodge star identifies bivectors with vectors:

In dimension $n$, however, the Hodge star sends bivectors to $(n-2)$-vectors. This construction produces a vector only when $n=3$. (cs.cmu.edu)

There is nevertheless a different seven-dimensional binary cross product, constructed by identifying $\mathbb R^7$ with the imaginary octonions:

It is bilinear and anticommutative and retains the perpendicularity and area-magnitude properties. Its symmetry structure differs from the three-dimensional case, and it does not satisfy all the same nested-product identities. Similarly, identifying $\mathbb R^3$ with imaginary quaternions recovers the ordinary cross product through the same half-commutator formula. (math.ucr.edu)

Historical development

The cross product’s algebraic background lies in nineteenth-century quaternion methods, which combined scalar and vector parts within a single multiplication. Modern vector notation separated dot and cross products into distinct operations. Josiah Willard Gibbs and Oliver Heaviside were central to this development and to its adoption in mathematical physics. Gibbs’s vector-analysis pamphlet was printed around 1881–1884; Heaviside received a copy in June 1888 and annotated it extensively. (math.ucr.edu)

References

1. 5 The Vector Productocw.mit.edu
2. World Web Math: Vector Calculus: Cross Productweb.mit.edu
3. 4 The Cross Product — Calculus Volume 3openstax.org
4. MIT 3.016 Fall 2005 — Lecture 11ocw.mit.edu
5. Geometry of Differential Forms — Pseudovectors and Pseudoscalarssites.science.oregonstate.edu
6. Discrete Differential Geometrycs.cmu.edu
7. The Octonionsmath.ucr.edu
8. Back to the Roots of Vector and Tensor Calculus. Heaviside versus Gibbsarxiv.org