The vector (or cross) product of the vectors a and b is defined to be:

a \times b = |a| |b| \sin \theta \hat{n}

where \theta is the angle between the vectors and \hat{n} is a unit vector perpendicular to both a and b.

The direction of \hat{n} is given by the right-hand rule - see the diagram.

The magnitude the cross product is equal to the area of the parallelogram that the vectors span.

If i, j and k are the familiar unit vectors, then:

i \times j = k, j \times k = i, k \times i = j and

i \times i = 0, j \times j = 0, k \times k = 0

If a = a_1i+a_2j+a_3k \quad and b = b_1i+b_2k+b_3k \quad then: a \times b = (a_2b_3-a_3b_2)i + (a_3b_1-a_1b_3)j + (a_1b_2-a_2b_1)k

a \times b = |a| |b| \sin \theta \hat{n}

where \theta is the angle between the vectors and \hat{n} is a unit vector perpendicular to both a and b.

The direction of \hat{n} is given by the right-hand rule - see the diagram.

The magnitude the cross product is equal to the area of the parallelogram that the vectors span.

If i, j and k are the familiar unit vectors, then:

i \times j = k, j \times k = i, k \times i = j and

i \times i = 0, j \times j = 0, k \times k = 0

If a = a_1i+a_2j+a_3k \quad and b = b_1i+b_2k+b_3k \quad then: a \times b = (a_2b_3-a_3b_2)i + (a_3b_1-a_1b_3)j + (a_1b_2-a_2b_1)k

## Software/Applets used on this page

## Glossary

### cross product

For vectors a and b, the cross product is the vector c whose magnitude is ab sin C, where C is the angle between the directions of the vectors, and which is perpendicular to both a and b.

### magnitude

A measure of the size of a mathematical object

### perpendicular

one line being at right angles to another.

### rule

A method for connecting one value with another.

### union

The union of two sets A and B is the set containing all the elements of A and B.

### unit vector

A vector with magnitude equal to 1.

### vector

A mathematical object with magnitude and direction.

## This question appears in the following syllabi:

Syllabus | Module | Section | Topic | Exam Year |
---|---|---|---|---|

AQA A-Level (UK - Pre-2017) | FP4 | Vectors | Cross product | - |

AQA A2 Further Maths 2017 | Pure Maths | Vectors | Cross Product | - |

AQA AS/A2 Further Maths 2017 | Pure Maths | Vectors | Cross Product | - |

CBSE XII (India) | Vectors and 3-D Geometry | Vectors | Cross product definition, geometrical interpretation, properties and application | - |

CCEA A-Level (NI) | FP3 | Vectors | Cross product | - |

Edexcel A-Level (UK - Pre-2017) | FP3 | Vectors | Cross product | - |

Edexcel AS Further Maths 2017 | Further Pure 1 | Vectors | Cross Product | - |

Edexcel AS/A2 Further Maths 2017 | Further Pure 1 | Vectors | Cross Product | - |

I.B. Higher Level | 4 | Vectors | Cross product | - |

Methods (UK) | M4 | Vectors | Cross product | - |

OCR A-Level (UK - Pre-2017) | FP3 | Vectors | Cross product | - |

OCR AS Further Maths 2017 | Pure Core | Vectors | Cross Product | - |

OCR MEI A2 Further Maths 2017 | Core Pure B | Vectors and 3-D Space | Cross Product | - |

OCR-MEI A-Level (UK - Pre-2017) | FP3 | Vectors | Cross product | - |

Pre-Calculus (US) | E1 | Vectors | Cross product | - |

Scottish Advanced Highers | M3 | Vectors | Cross product | - |

Scottish (Highers + Advanced) | AM3 | Vectors | Cross product | - |

Universal (all site questions) | V | Vectors | Cross product | - |