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Vectors AB and AC are added using the "parallelogram law" to produce vector AD.


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.

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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.


A rule that connects one value in one set with one and only one value in another set.


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

unit vector

A vector with magnitude equal to 1.


A mathematical object with magnitude and direction.

Full Glossary List

This question appears in the following syllabi:

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
J VectorsJ1 Vectors in Two Dimensions2D 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