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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.

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.

Full Glossary List

## This question appears in the following syllabi:

SyllabusModuleSectionTopicExam Year
AQA A-Level (UK - Pre-2017)C4Vectors2D Vector geometry-
AQA AS Maths 2017MechanicsVectorsVector Basics-
AQA AS/A2 Maths 2017MechanicsVectorsVector Basics-
CBSE XII (India)Vectors and 3-D GeometryVectorsVectors and scalars, magnitude and direction of a vector-
CCEA A-Level (NI)C4Vectors2D Vector geometry-
CIE A-Level (UK)P1Vectors2D Vector geometry-
Edexcel A-Level (UK - Pre-2017)C4Vectors2D Vector geometry-
Edexcel AS Maths 2017Pure MathsVectorsVector Basics-
Edexcel AS/A2 Maths 2017Pure MathsVectorsVector Basics-
I.B. Higher Level4Vectors2D Vector geometry-
I.B. Standard Level4Vectors2D Vector geometry-
Methods (UK)M4Vectors2D Vector geometry-
OCR A-Level (UK - Pre-2017)C4Vectors2D Vector geometry-
OCR AS Maths 2017Pure MathsVectorsVector Basics-
OCR MEI AS Maths 2017Pure MathsVectorsVector Basics-
OCR-MEI A-Level (UK - Pre-2017)C4Vectors2D Vector geometry-
Pre-Calculus (US)E1Vectors2D Vector geometry-
Pre-U A-Level (UK)6Vectors2D Vector geometry-
Scottish (Highers + Advanced)HM3Vectors2D Vector geometry-
Scottish HighersM3Vectors2D Vector geometry-
Universal (all site questions)VVectors2D Vector geometry-
WJEC A-Level (Wales)C4Vectors2D Vector geometry-