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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.


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.

Software/Applets used on this page

This page uses JSXGraph

JSXGraph is a cross-browser library for interactive geometry, function plotting, charting, and data visualization in a web browser. It is implemented completely in JavaScript, does not rely on any other library. It uses SVG and VML and is fully HTML5 compliant.

This page also uses the MathJax system for displaying maths symbols.


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