# Definition:Vector Cross Product

*This page is about Cross Product in the context of Vector Algebra. For other uses, see Cross Product.*

## Definition

Let $\mathbf a$ and $\mathbf b$ be vectors in a vector space $\mathbf V$ of $3$ dimensions:

- $\mathbf a = a_i \mathbf i + a_j \mathbf j + a_k \mathbf k$
- $\mathbf b = b_i \mathbf i + b_j \mathbf j + b_k \mathbf k$

where $\tuple {\mathbf i, \mathbf j, \mathbf k}$ is the standard ordered basis of $\mathbf V$.

### Definition 1

The **vector cross product**, denoted $\mathbf a \times \mathbf b$, is defined as:

- $\mathbf a \times \mathbf b = \begin{vmatrix} \mathbf i & \mathbf j & \mathbf k\\ a_i & a_j & a_k \\ b_i & b_j & b_k \\ \end{vmatrix}$

where $\begin {vmatrix} \ldots \end {vmatrix}$ is interpreted as a determinant.

More directly:

- $\mathbf a \times \mathbf b = \paren {a_j b_k - a_k b_j} \mathbf i - \paren {a_i b_k - a_k b_i} \mathbf j + \paren {a_i b_j - a_j b_i} \mathbf k$

### Definition 2

The **vector cross product**, denoted $\mathbf a \times \mathbf b$, is defined as:

- $\mathbf a \times \mathbf b = \norm {\mathbf a} \norm {\mathbf b} \sin \theta \, \mathbf {\hat n}$

where:

- $\norm {\mathbf a}$ denotes the length of $\mathbf a$
- $\theta$ denotes the angle from $\mathbf a$ to $\mathbf b$, measured in the positive direction
- $\hat {\mathbf n}$ is the unit vector perpendicular to both $\mathbf a$ and $\mathbf b$ in the direction according to the right-hand rule.

## Complex Numbers

The definition is slightly different when the vector space under consideration is the complex plane, as the latter is of $2$ dimensions.

This is a modification of the vector cross product in which the resulting product is taken to be the length of the hypothetical vector which would be considered as being perpendicular to the complex plane.

Let $z_1 := x_1 + i y_1$ and $z_2 := x_2 + i y_2$ be complex numbers.

### Definition 1

The **cross product** of $z_1$ and $z_2$ is defined as:

- $z_1 \times z_2 = x_1 y_2 - y_1 x_2$

### Definition 2

The **cross product** of $z_1$ and $z_2$ is defined as:

- $z_1 \times z_2 = \cmod {z_1} \, \cmod {z_2} \sin \theta$

where:

- $\cmod {z_1}$ denotes the complex modulus of $z_1$
- $\theta$ denotes the angle from $z_1$ to $z_2$, measured in the positive direction.

### Definition 3

The **cross product** of $z_1$ and $z_2$ is defined as:

- $z_1 \times z_2 := \map \Im {\overline {z_1} z_2}$

where:

- $\map \Im z$ denotes the imaginary part of a complex number $z$
- $\overline {z_1}$ denotes the complex conjugate of $z_1$
- $\overline {z_1} z_2$ denotes complex multiplication.

### Definition 4

The **cross product** of $z_1$ and $z_2$ is defined as:

- $z_1 \times z_2 := \dfrac {\overline {z_1} z_2 - z_1 \overline {z_2}} {2 i}$

where:

- $\overline {z_1}$ denotes the complex conjugate of $z_1$
- $\overline {z_1} z_2$ denotes complex multiplication.

## Also known as

The **vector cross product** is often called just the **cross product** when there is no chance of confusion with other types of cross product.

The term **vector product** can also sometimes be seen, but again this can be ambiguous.

## Examples

### Couple Exerted by Force

Let $\mathbf F$ be a force acting at a point $P$ on a body $B$ axially about an axis of rotation $R$ such that the distance from $P$ to $R$ is represented by the displacement vector $\mathbf d$.

Then the couple exerted on $B$ by $\mathbf F$ is defined as:

- $\mathbf T = \mathbf F \times \mathbf d = \norm {\mathbf F} \norm {\mathbf d} \mathbf {\hat t} \sin \theta$

where:

- $\times$ denotes
**vector cross product** - $\mathbf {\hat t}$ denotes the unit vector perpendicular to both $\mathbf F$ and $\mathbf d$ according to the right-hand rule
- $\theta$ is the angle between the directions of $\mathbf F$ and $\mathbf d$.

## Also see

- Lagrange's Formula
- Vector Cross Product Distributes over Addition
- Vector Cross Product is Anticommutative
- Vector Cross Product is not Associative
- Definition:Dot Product

- Results about
**Vector Cross Product**can be found here.

## Historical Note

During the course of development of vector analysis, various notations for the vector cross product were introduced, as follows:

Symbol | Used by |
---|---|

$\mathbf a \times \mathbf b$ | Josiah Willard Gibbs and Edwin Bidwell Wilson |

$V \mathbf a \mathbf b$ | Oliver Heaviside |

$\sqbrk {\mathscr A \mathscr B}$ | Max Abraham |

$\sqbrk {\mathfrak A \mathfrak B}$ | Vladimir Sergeyevitch Ignatowski |

$\sqbrk {\mathbf A \cdot \mathbf B}$ | Hendrik Antoon Lorentz |

$\mathbf a \wedge \mathbf b$ | Cesare Burali-Forti and Roberto Marcolongo |

## Technical Note

The $\LaTeX$ code for \(\mathbf A \times \mathbf B\) is `\mathbf A \times \mathbf B`

.

The $\LaTeX$ code for \(\mathbf A \wedge \mathbf B\) is `\mathbf A \wedge \mathbf B`

.

In this context, $\wedge$ is usually referred to as **wedge**.

## Sources

- 1964: D.E. Rutherford:
*Classical Mechanics*(3rd ed.) ... (previous) ... (next): Introduction - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem