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Cross product (vector product)

The cross product is a binary operation on two three-dimensional vectors that yields a third vector perpendicular to both, with magnitude equal to the parallelogram area; widely used in physics and geometry.

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Overview

The cross product, also called the vector product, is an operation defined for two vectors in three-dimensional space. Given two nonparallel vectors it returns a third vector that is perpendicular to both inputs and whose magnitude equals the area of the parallelogram they span. It is most commonly applied to three-dimensional vectors and is frequently introduced alongside the dot product in courses on calculus, linear algebra, and physics.

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Cross product
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Definition and coordinate formula

If a = (a1, a2, a3) and b = (b1, b2, b3) are vectors expressed in a right-handed Cartesian coordinate system, their cross product a × b is the vector

  • (a2 b3 − a3 b2, a3 b1 − a1 b3, a1 b2 − a2 b1).
This can be remembered using the determinant of a 3×3 matrix with the unit vectors i, j, k in the top row. The direction of a × b follows the right-hand rule: curling the fingers from a toward b makes the thumb point in the direction of the product.

Key properties

The cross product has several standard algebraic and geometric properties:

  • Anticommutative: a × b = −(b × a).
  • Distributive over addition: a × (b + c) = a × b + a × c.
  • Scalar multiplication: (λa) × b = λ(a × b) = a × (λb).
  • Perpendicularity: a × b is orthogonal to both a and b.
  • Magnitude: |a × b| = |a| |b| sin θ, where θ is the angle between a and b; this equals the area of the parallelogram formed by a and b.

Applications and examples

The cross product appears in many practical contexts. Examples include:

  • Physics: torque τ = r × F, angular momentum L = r × p, and the magnetic Lorentz force F = q(v × B).
  • Geometry and engineering: computing a normal vector to a surface from two tangent directions, or determining the orientation of a triangle in 3D modeling.
  • Area and volume checks: the magnitude gives areas; pairing with a third vector yields the scalar triple product (a × b) · c, which equals the volume of the parallelepiped spanned by a, b, and c.

Generalizations and distinctions

The standard cross product as a binary vector-valued operation is special to three-dimensional Euclidean space (and has a related construction in seven dimensions that is less commonly used). More generally, exterior algebra provides tools such as the wedge product for higher dimensions; the Hodge star operator then converts bivectors into vectors in spaces with a chosen metric and orientation. A common point of comparison is the dot product, which produces a scalar and measures projection rather than perpendicular magnitude.

Historical and practical notes

The vector cross product emerged as part of nineteenth-century developments in vector calculus and classical mechanics. Today it is a standard tool in physics, computer graphics, robotics, and engineering because it encodes both orientation and area information compactly. When using it, be mindful of coordinate handedness (right- vs left-handed systems) because the sign of the result depends on that convention.

Cross product
Cross product

Geometric definition

The cross product \vec{a}\times\vec{b} of two vectors {\vec {a}}and {\vec {b}}in three-dimensional visual space is a vector orthogonal to {\vec {a}}and {\vec {b}}, and hence orthogonal to the {\vec {b}}plane spanned by {\vec {a}}and

This vector is oriented such that {\vec {a}},{\vec {b}}and \vec{a}\times\vec{b} form a right system in that order. Mathematically, this means that the three vectors {\vec {a}},{\vec {b}}and \vec{a}\times\vec{b}are oriented the same as the vectors \vec e_1, \vec e_2and \vec e_3the standard basis. In physical space, it means that they behave like the thumb, index finger and splayed middle finger of the right hand (right-hand rule). Rotating the first vector {\vec {a}}into the second vector {\vec {b}}yields the positive direction of the vector \vec{a}\times\vec{b}via the right-hand screw sense.

The magnitude of \vec{a}\times\vec{b}gives the area of the parallelogram {\vec {b}}spanned by {\vec {a}}and . Expressed by the angle θ {\vec {a}}{\vec {b}}enclosed by and \theta holds that

|\vec{a}\times\vec{b}| = |\vec{a}|\, |\vec{b}|\, \sin\theta \,.

Where \vert\vec{a}\vertand \vert\vec{b}\vert denote the lengths of the vectors {\vec {a}} and \vec{b}, and \sin \theta\,is the sine of the angle θ enclosed by them \theta .

In summary

\vec{a}\times\vec{b} = (|\vec{a}|\, |\vec{b}|\, \sin\theta) \, \vec{n}\,,

where the vector {\vec {n}} is the unit vector \vec{b}perpendicular to {\vec {a}}and that completes them to a right system.

Right hand rule
Right hand rule

Spellings

Depending on the country, different notations are used for the vector product. In English and German-speaking countries, the vector product of two vectors {\vec {a}}and is \vec{b}usually written as \vec{a}\times\vec{b}, whereas in France and Italy the notation \vec{a}\wedge\vec{b}preferred. In Russia, the vector product is often {\displaystyle [{\vec {a}},{\vec {b}}]}notated as {\displaystyle [{\vec {a}}\ {\vec {b}}]}or .

The notation \vec{a}\wedge\vec{b}and the term outer product are used not only for the vector product, but also for the conjunction that assigns a so-called bivector to two vectors, see Graßmann algebra.

Questions and answers

Q: What is the cross product?

A: The cross product is a mathematical operation that can be done between two three-dimensional vectors.

Q: How is the cross product often represented?

A: The cross product is often represented by the symbol × or \times.

Q: What happens after performing the cross product?

A: After performing the cross product, a new vector is formed.

Q: What is the relationship between the cross product vector and the vectors that were "crossed"?

A: The cross product of two vectors is always perpendicular (it makes a corner-shaped angle) to both of the vectors which were "crossed".

Q: In what dimension does the cross product normally work?

A: Cross product normally works only in three-dimensional space.

Q: What are the three dimensions where cross product can be performed?

A: The three dimensions where cross product can be performed are up or down, left or right, and forward or backwards.

Q: Why can cross product normally only work in three-dimensional space?

A: Cross product normally works only in three-dimensional space because those are the dimensions where you can go up or down, left or right, and forward or backwards.

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URL: https://en.alegsaonline.com/art/24329

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