The vector subtraction AB - AC is illustrated using the "parallelogram law" to produce vector AD.

## Summary/Background

Vectors can be added together (vector addition), subtracted (vector subtraction) and multiplied by scalars (scalar multiplication). Vector multiplication is not uniquely defined, but a number of different types of products, such as the dot product and cross product can be defined for pairs of vectors.

A vector from a point A to a point B is denoted \vec{AB} , and a vector v may be denoted \bar{v} . The point A is often called the "tail" of the vector, and B is called the vector's "head." A vector with unit length is called a unit vector and often denoted using a hat, \hat{v} .

Vectors were born in the first two decades of the 19th century with the geometric representations of complex numbers. Caspar Wessel (1745--1818), Jean Robert Argand (1768--1822), Carl Friedrich Gauss (1777--1855), and at least one or two others conceived of complex numbers as points in the two-dimensional plane, i.e., as two-dimensional vectors. Mathematicians and scientists worked with and applied these new numbers in various ways; for example, Gauss made crucial use of complex numbers to prove the Fundamental Theorem of Algebra (1799). In 1837, William Rowan Hamilton (1805-1865) showed that the complex numbers could be considered abstractly as ordered pairs (a, b) of real numbers. This idea was a part of the campaign of many mathematicians, including Hamilton himself, to search for a way to extend the two-dimensional "numbers" to three dimensions; but no one was able to accomplish this, while preserving the basic algebraic properties of real and complex numbers.

A vector from a point A to a point B is denoted \vec{AB} , and a vector v may be denoted \bar{v} . The point A is often called the "tail" of the vector, and B is called the vector's "head." A vector with unit length is called a unit vector and often denoted using a hat, \hat{v} .

Vectors were born in the first two decades of the 19th century with the geometric representations of complex numbers. Caspar Wessel (1745--1818), Jean Robert Argand (1768--1822), Carl Friedrich Gauss (1777--1855), and at least one or two others conceived of complex numbers as points in the two-dimensional plane, i.e., as two-dimensional vectors. Mathematicians and scientists worked with and applied these new numbers in various ways; for example, Gauss made crucial use of complex numbers to prove the Fundamental Theorem of Algebra (1799). In 1837, William Rowan Hamilton (1805-1865) showed that the complex numbers could be considered abstractly as ordered pairs (a, b) of real numbers. This idea was a part of the campaign of many mathematicians, including Hamilton himself, to search for a way to extend the two-dimensional "numbers" to three dimensions; but no one was able to accomplish this, while preserving the basic algebraic properties of real and complex numbers.

## Software/Applets used on this page

Uses the CabriJava interactive geometry applet. You can download the file by double-clicking on the display and then selecting the option at bottom right. You can then load it into your copy of Cabri.

## 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.

### dot product

For vectors a and b, a.b=|a||b|cos C, where C is the angle between the directions of the vectors.

### geometric

A sequence where each term is obtained by multiplying the previous one by a constant.

### 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) | C4 | Vectors | 2D Vector geometry | - |

AQA AS Maths 2017 | Mechanics | Vectors | Vector Basics | - |

AQA AS/A2 Maths 2017 | Mechanics | Vectors | Vector Basics | - |

CBSE XII (India) | Vectors and 3-D Geometry | Vectors | Vectors and scalars, magnitude and direction of a vector | - |

CCEA A-Level (NI) | C4 | Vectors | 2D Vector geometry | - |

CIE A-Level (UK) | P1 | Vectors | 2D Vector geometry | - |

Edexcel A-Level (UK - Pre-2017) | C4 | Vectors | 2D Vector geometry | - |

Edexcel AS Maths 2017 | Pure Maths | Vectors | Vector Basics | - |

Edexcel AS/A2 Maths 2017 | Pure Maths | Vectors | Vector Basics | - |

I.B. Higher Level | 4 | Vectors | 2D Vector geometry | - |

I.B. Standard Level | 4 | Vectors | 2D Vector geometry | - |

Methods (UK) | M4 | Vectors | 2D Vector geometry | - |

OCR A-Level (UK - Pre-2017) | C4 | Vectors | 2D Vector geometry | - |

OCR AS Maths 2017 | Pure Maths | Vectors | Vector Basics | - |

OCR MEI AS Maths 2017 | Pure Maths | Vectors | Vector Basics | - |

OCR-MEI A-Level (UK - Pre-2017) | C4 | Vectors | 2D Vector geometry | - |

Pre-Calculus (US) | E1 | Vectors | 2D Vector geometry | - |

Pre-U A-Level (UK) | 6 | Vectors | 2D Vector geometry | - |

Scottish (Highers + Advanced) | HM3 | Vectors | 2D Vector geometry | - |

Scottish Highers | M3 | Vectors | 2D Vector geometry | - |

Universal (all site questions) | V | Vectors | 2D Vector geometry | - |

WJEC A-Level (Wales) | C4 | Vectors | 2D Vector geometry | - |