# Cross product

The cross product, also called the vector product, is an operation on two vectors in three-dimensional space that produces a third vector perpendicular to both inputs, with magnitude equal to the product of their lengths and the sine of the angle between them, and with direction fixed by the right-hand rule.<sup>[1](https://openstax.org/books/calculus-volume-3/pages/2-4-the-cross-product)</sup><sup> • </sup><sup>[2](https://mathworld.wolfram.com/CrossProduct.html)</sup> It is one of the two standard ways of multiplying vectors, the other being the dot product, which returns a scalar instead of a vector.<sup>[3](https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-8-vectors)</sup>

| Key fact | Detail |
|---|---|
| Definition | For vectors \( \mathbf{u} \) and \( \mathbf{v} \) with angle \( \theta \) between them, \( \|\mathbf{u} \times \mathbf{v}\| = \|\mathbf{u}\| \, \|\mathbf{v}\| \sin \theta \), directed perpendicular to both by the right-hand rule<sup>[1](https://openstax.org/books/calculus-volume-3/pages/2-4-the-cross-product)</sup><sup> • </sup><sup>[2](https://mathworld.wolfram.com/CrossProduct.html)</sup> |
| Component formula | \( \mathbf{u} \times \mathbf{v} = (u_2 v_3 - u_3 v_2,\; u_3 v_1 - u_1 v_3,\; u_1 v_2 - u_2 v_1) \), computed as a 3×3 determinant with the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) in the first row<sup>[2](https://mathworld.wolfram.com/CrossProduct.html)</sup> |
| Anticommutativity | \( \mathbf{u} \times \mathbf{v} = -\mathbf{v} \times \mathbf{u} \); swapping the order reverses the direction<sup>[4](https://www.feynmanlectures.caltech.edu/I_20.html)</sup> |
| Geometric meaning | The magnitude equals the area of the parallelogram spanned by the two vectors; the triple scalar product \( \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) \) gives the volume of the parallelepiped they span<sup>[1](https://openstax.org/books/calculus-volume-3/pages/2-4-the-cross-product)</sup> |
| Dimensional scope | The usual vector cross product is defined in three dimensions, and a seven-dimensional analog also exists; a binary product with the cross product's basic properties on \( \mathbf{R}^n \) exists only for \( n = 0, 1, 3, \) or \( 7 \)<sup>[1](https://openstax.org/books/calculus-volume-3/pages/2-4-the-cross-product)</sup><sup> • </sup><sup>[5](https://arxiv.org/abs/1212.3515)</sup> |
| Physics role | Torque is \( \boldsymbol{\tau} = \mathbf{r} \times \mathbf{F} \) and angular momentum is \( \mathbf{L} = \mathbf{r} \times \mathbf{p} \); both are pseudovectors<sup>[4](https://www.feynmanlectures.caltech.edu/I_20.html)</sup> |
| Origin | Extracted from William Rowan Hamilton's quaternions by Josiah Willard Gibbs and Oliver Heaviside in the 1880s<sup>[6](https://www.scielo.br/j/rbef/a/XVRqzXNzJHmwDKsQbPqHMFw/abstract/?lang=en)</sup> |

## Definition and magnitude

For two vectors \( \mathbf{u} \) and \( \mathbf{v} \) in \( \mathbf{R}^3 \) with \( \theta \) the angle between them, the cross product \( \mathbf{u} \times \mathbf{v} \) is the vector whose length is

\[ \|\mathbf{u} \times \mathbf{v}\| = \|\mathbf{u}\| \, \|\mathbf{v}\| \sin \theta, \]

whose direction is perpendicular to both \( \mathbf{u} \) and \( \mathbf{v} \), and whose sense is fixed by the right-hand rule in a positively oriented basis.<sup>[1](https://openstax.org/books/calculus-volume-3/pages/2-4-the-cross-product)</sup><sup> • </sup><sup>[2](https://mathworld.wolfram.com/CrossProduct.html)</sup> Because \( \sin \theta = 0 \) when the vectors are parallel, the cross product of any vector with itself is the zero vector.<sup>[4](https://www.feynmanlectures.caltech.edu/I_20.html)</sup>

The magnitude has a direct geometric reading: it is the area of the parallelogram whose sides are the two vectors.<sup>[1](https://openstax.org/books/calculus-volume-3/pages/2-4-the-cross-product)</sup> The identity behind this, \( \|\mathbf{u} \times \mathbf{v}\|^2 + (\mathbf{u} \cdot \mathbf{v})^2 = \|\mathbf{u}\|^2 \|\mathbf{v}\|^2 \), is the Pythagorean property used in the standard proofs.<sup>[5](https://arxiv.org/abs/1212.3515)</sup>

The usual cross product is defined for vectors in three dimensions, and a seven-dimensional analog also exists.<sup>[5](https://arxiv.org/abs/1212.3515)</sup> In two dimensions it is impossible to generate a vector simultaneously orthogonal to two nonparallel vectors, since no such direction exists in the plane.<sup>[1](https://openstax.org/books/calculus-volume-3/pages/2-4-the-cross-product)</sup>

## Component formula and determinant notation

Writing \( \mathbf{u} = \langle u_1, u_2, u_3 \rangle \) and \( \mathbf{v} = \langle v_1, v_2, v_3 \rangle \), the cross product is

\[ \mathbf{u} \times \mathbf{v} = (u_2 v_3 - u_3 v_2)\,\mathbf{i} + (u_3 v_1 - u_1 v_3)\,\mathbf{j} + (u_1 v_2 - u_2 v_1)\,\mathbf{k}. \]

This follows from the products of the standard unit vectors: \( \mathbf{i} \times \mathbf{j} = \mathbf{k} \), \( \mathbf{j} \times \mathbf{k} = \mathbf{i} \), and \( \mathbf{k} \times \mathbf{i} = \mathbf{j} \), while reversing the order of any pair gives the negative, and \( \mathbf{i} \times \mathbf{i} = \mathbf{j} \times \mathbf{j} = \mathbf{k} \times \mathbf{k} = \mathbf{0} \).<sup>[1](https://openstax.org/books/calculus-volume-3/pages/2-4-the-cross-product)</sup><sup> • </sup><sup>[7](https://openstax.org/books/university-physics-volume-1/pages/2-4-products-of-vectors)</sup> When the two unit vectors appear in cyclic order the result is the remaining unit vector; in any other order it is the antiparallel unit vector.<sup>[7](https://openstax.org/books/university-physics-volume-1/pages/2-4-products-of-vectors)</sup>

The component formula is awkward to memorize, so it is usually written as a determinant with the unit vectors across the first row and the components of \( \mathbf{u} \) and \( \mathbf{v} \) in the second and third rows; expanding along the first row reproduces the formula.<sup>[1](https://openstax.org/books/calculus-volume-3/pages/2-4-the-cross-product)</sup><sup> • </sup><sup>[2](https://mathworld.wolfram.com/CrossProduct.html)</sup> In two dimensions there is a scalar analog: for \( \mathbf{u} = (u_x, u_y) \) and \( \mathbf{v} = (v_x, v_y) \), the quantity \( u_x v_y - u_y v_x \) is the signed area of the parallelogram they span.<sup>[2](https://mathworld.wolfram.com/CrossProduct.html)</sup>

## Algebraic properties

The cross product is bilinear and anticommutative: \( \mathbf{u} \times \mathbf{v} = -\mathbf{v} \times \mathbf{u} \).<sup>[4](https://www.feynmanlectures.caltech.edu/I_20.html)</sup> It is not associative, and the standard identities collected by Richard Feynman in his lecture notes include the scalar triple product symmetry \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} \) and the vector triple product expansion

\[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \mathbf{b}\,(\mathbf{a} \cdot \mathbf{c}) - \mathbf{c}\,(\mathbf{a} \cdot \mathbf{b}). \]

Also \( \mathbf{a} \cdot (\mathbf{a} \times \mathbf{b}) = 0 \), which restates orthogonality.<sup>[4](https://www.feynmanlectures.caltech.edu/I_20.html)</sup>

The triple scalar product \( \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) \) gives the volume of the parallelepiped formed by the three vectors, and it equals zero exactly when the vectors are coplanar.<sup>[1](https://openstax.org/books/calculus-volume-3/pages/2-4-the-cross-product)</sup>

## Role in physics

The cross product enters mechanics wherever rotation appears. The torque of a force \( \mathbf{F} \) applied at position \( \mathbf{r} \) is \( \boldsymbol{\tau} = \mathbf{r} \times \mathbf{F} \), and the angular momentum of a particle with momentum \( \mathbf{p} \) is \( \mathbf{L} = \mathbf{r} \times \mathbf{p} \). The rotational analog of Newton's second law is \( \boldsymbol{\tau} = d\mathbf{L}/dt \), and when the total external torque is zero the total angular momentum is conserved.<sup>[4](https://www.feynmanlectures.caltech.edu/I_20.html)</sup> The Yale physicist Ramamurti Shankar, in his open mechanics course, derives \( d\mathbf{L}/dt = \boldsymbol{\tau} \) by differentiating \( \mathbf{r} \times \mathbf{p} \); the term \( \mathbf{v} \times \mathbf{p} \) vanishes because velocity and momentum are parallel, and parallel vectors have zero cross product.<sup>[8](https://oyc.yale.edu/physics/phys-200/lecture-11)</sup>

Quantities defined with one cross product, such as torque, angular momentum, angular velocity, and the magnetic field, are called axial vectors or [pseudovectors](https://www.edgepedia.org/pseudovector); they transform differently from ordinary polar vectors under reflection.<sup>[4](https://www.feynmanlectures.caltech.edu/I_20.html)</sup> This is a consequence of the operation's dependence on handedness: the cross product does not respect reflection rules, so its outcome depends on the orientation convention chosen for the coordinate system.<sup>[9](https://arxiv.org/abs/1408.5799)</sup>

## History

The cross product descends from quaternion multiplication. [William Rowan Hamilton](https://www.edgepedia.org/william-rowan-hamilton) regarded a quaternion \( a + bi + cj + dk \) as a scalar part plus a vector part and handled vectors within that combined system.<sup>[10](https://www.nationalacademies.org/read/11540/chapter/13)</sup> In the imaginary quaternions, the product \( \tfrac{1}{2}(ab - ba) \) reproduces exactly the usual cross product.<sup>[11](https://math.ucr.edu/home/baez/week169.html)</sup>

In the 1880s [Josiah Willard Gibbs](https://www.edgepedia.org/josiah-willard-gibbs), a physicist at Yale, and the English physicist [Oliver Heaviside](https://www.edgepedia.org/oliver-heaviside) separated the scalar and vector parts of the quaternion product, treated them as independent operations, and founded modern vector analysis.<sup>[10](https://www.nationalacademies.org/read/11540/chapter/13)</sup><sup> • </sup><sup>[6](https://www.scielo.br/j/rbef/a/XVRqzXNzJHmwDKsQbPqHMFw/abstract/?lang=en)</sup> Gibbs printed his *Elements of Vector Analysis* in New Haven in 1881–1884 for use only in his own classes, and he and Heaviside used the new methods to express Maxwell's electromagnetic laws in the concise form now called [Maxwell's equations](https://www.edgepedia.org/maxwells-equations).<sup>[12](https://library.si.edu/digital-library/book/elementsvectora00gibb)</sup> The notation \( \mathbf{a} \times \mathbf{b} \) is due to Gibbs and Edwin Bidwell Wilson, while Heaviside wrote \( V\mathbf{a}\mathbf{b} \).<sup>[13](https://proofwiki.org/wiki/Definition%3AVector_Cross_Product/Historical_Note)</sup> The change was contested: an 1893 *Nature* report described the dispute between Gibbs's followers and the quaternionists, with Heaviside supporting Gibbs and calling the quaternion a quantity no physicist wants.<sup>[14](https://www.nature.com/articles/047590b0)</sup> By 1910 the Gibbs–Heaviside system had established itself as the dominant vector analysis in mathematics, physics, and engineering.<sup>[15](https://www.math.ucdavis.edu/~temple/MAT21D/SUPPLEMENTARY-ARTICLES/Crowe_History-of-Vectors.pdf)</sup>

## Higher dimensions

A natural question is whether the cross product extends to \( \mathbf{R}^n \) for \( n > 3 \). If a binary product on \( \mathbf{R}^n \) is required to satisfy the perpendicular, Pythagorean, and bilinear properties of the three-dimensional cross product, such a product exists only for \( n = 0, 1, 3, \) or \( 7 \).<sup>[5](https://arxiv.org/abs/1212.3515)</sup> [Beno Eckmann](https://www.edgepedia.org/beno-eckmann) gave the first proof in 1943 using algebraic topology, under the weaker assumption of continuity.<sup>[5](https://arxiv.org/abs/1212.3515)</sup> The connection with dimensions 3 and 7 runs through the normed division algebras: the imaginary quaternions carry the three-dimensional cross product and the imaginary octonions a seven-dimensional analog, both given by \( a \times b = \tfrac{1}{2}(ab - ba) \).<sup>[11](https://math.ucr.edu/home/baez/week169.html)</sup> The seven-dimensional product is bilinear and anticommutative and satisfies the same norm identity, but it does not make the imaginary octonions a [Lie algebra](https://www.edgepedia.org/lie-algebra), unlike the three-dimensional case.<sup>[16](https://math.ucr.edu/home/baez/octonions/node14.html)</sup>

In any dimension \( n \), a determinant can define a product of \( n - 1 \) vectors returning one vector orthogonal to all inputs, and the exterior algebra's wedge product provides the general framework into which the three-dimensional cross product fits; the ordinary cross product is the Hodge dual of the wedge product of two vectors in \( \mathbf{R}^3 \).<sup>[17](https://mathworld.wolfram.com/WedgeProduct.html)</sup> In physics, quantities such as angular velocity and the magnetic field can instead be treated as antisymmetric second-rank tensors, which yields forms of the cross product and curl equations that generalize to higher dimensions.<sup>[18](https://arxiv.org/abs/hep-ph/0609260)</sup>

## References

1. [2.4 The Cross Product, Calculus Volume 3, OpenStax](https://openstax.org/books/calculus-volume-3/pages/2-4-the-cross-product)
2. [Cross Product, Wolfram MathWorld](https://mathworld.wolfram.com/CrossProduct.html)
3. [10.8 Vectors, Algebra and Trigonometry 2e, OpenStax](https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-8-vectors)
4. [The Feynman Lectures on Physics, Vol. I Ch. 20: Rotation in Space](https://www.feynmanlectures.caltech.edu/I_20.html)
5. [Peter F. McLoughlin (2012). When Does a Cross Product on R^n Exist? arXiv](https://arxiv.org/abs/1212.3515)
6. [On the origins of the scalar and vectorial product definitions, Revista Brasileira de Ensino de Física 31 (2), 2009](https://www.scielo.br/j/rbef/a/XVRqzXNzJHmwDKsQbPqHMFw/abstract/?lang=en)
7. [2.4 Products of Vectors, University Physics Volume 1, OpenStax](https://openstax.org/books/university-physics-volume-1/pages/2-4-products-of-vectors)
8. [PHYS 200 Lecture 11: Torque, Open Yale Courses (Ramamurti Shankar)](https://oyc.yale.edu/physics/phys-200/lecture-11)
9. [Cross product in N Dimensions – the doublewedge product, arXiv](https://arxiv.org/abs/1408.5799)
10. [John Derbyshire, Unknown Quantity: A Real and Imaginary History of Algebra, chapter 8, Joseph Henry Press](https://www.nationalacademies.org/read/11540/chapter/13)
11. [John Baez, This Week's Finds 169: vector product algebras](https://math.ucr.edu/home/baez/week169.html)
12. [J. Willard Gibbs, Elements of Vector Analysis (1881–1884), Smithsonian Libraries](https://library.si.edu/digital-library/book/elementsvectora00gibb)
13. [Definition: Vector Cross Product, Historical Note, ProofWiki](https://proofwiki.org/wiki/Definition%3AVector_Cross_Product/Historical_Note)
14. [Recent Innovations in Vector Theory, Nature 47, 590–593 (1893)](https://www.nature.com/articles/047590b0)
15. [Michael J. Crowe, A History of Vector Analysis (excerpt), UC Davis](https://www.math.ucdavis.edu/~temple/MAT21D/SUPPLEMENTARY-ARTICLES/Crowe_History-of-Vectors.pdf)
16. [John Baez, The Octonions, section on the seven-dimensional cross product](https://math.ucr.edu/home/baez/octonions/node14.html)
17. [Wedge Product, Wolfram MathWorld](https://mathworld.wolfram.com/WedgeProduct.html)
18. [Generalizing Cross Products and Maxwell's Equations to Universal Extra Dimensions, arXiv](https://arxiv.org/abs/hep-ph/0609260)

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