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The vectors i and j are orthogonal unit vectors. Vector OA is a position vector and vector BC is a displacement vector.

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.

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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.

displacement

The distance and direction an object moves between its starting and finishing positions. It is a vector quantity.

dot product

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

function

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

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:

SyllabusModuleSectionTopic
AQA A-Level (UK - Pre-2017)C4Vectors2D Vector geometry
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
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-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