The concept of a **vector** is fundamental in physics and engineering. Although the word now has many meanings (see also vector, and generalizations below), its original and most common meaning in those fields is a quantity that has a close relationship to spatial directions. The use of *vector* in this article refers to that original meaning, except where otherwise noted.

Often informally described as an object with a "magnitude" (size) and "direction", a vector is more formally defined by its relationship to the spatial coordinate system under rotations. Alternatively, it can be defined in a coordinate-free fashion via a tangent space of a three-dimensional manifold in the language of differential geometry. These definitions are discussed in more detail below.

Such a vector is a special case of a tensor and is also analogous to a four-vector in relativity (and is sometimes therefore called a *three-vector* in reference to the three spatial dimensions, although this term also has another meaning for p-vectors of differential geometry). Vectors are the building blocks of vector fields and vector calculus.

Table of contents |

2 Representation of a vector 3 Vector Equality 4 Vector Addition and Subtraction 5 Dot Product 6 Cross Product 7 Scalar Triple Product 8 External links |

The notion of having a "magnitude" and "direction" is formalized by saying that the vector has components that transform like the coordinates under rotations. That is, if the coordinate system undergoes a rotation described by a rotation matrix *R*, so that a coordinate vector **x** is transformed to **x**' = *R***x**, then any other vector **v** is similarly transformed via **v**' = *R***v**. More generally, a vector is a tensor of contravariant rank one. In differential geometry, the term *vector* usually refers to quantities that are closely related to tangent spaces of a differentiable manifold (assumed to be three-dimensional and equipped with a positive definite Riemannian metric). (A four-vector is a related concept when dealing with a 4 dimensional spacetime manifold in relativity.)

Examples of vectors include displacement, velocity, electric field, momentum, force, and acceleration.

Vectors can be contrasted with scalar quantities such as distance, speed, energy, time, temperature, charge, power, work, and mass, which have magnitude, but no direction (they are invariant under coordinate rotations). The magnitude of any vector is a scalar.

A related concept is that of a pseudovector (or **axial vector**). This is a quantity that transforms like a vector under proper rotations, but gains an additional sign flip under improper rotations. Examples of pseudovectors include magnetic field, torque, and angular momentum. (This distinction between vectors and pseudovectors is often ignored, but it becomes important in studying symmetry properties.) To distinguish from pseudo/axial vectors, an ordinary vector is sometimes called a **polar vector**.

Sometimes, one speaks informally of *bound* or *fixed* vectors, which are vectors additionally characterized by a "base point". Most often, this term is used for position vectors (relative to an origin point). More generally, however, the physical interpretation of a particular vector can be parameterized by any number of quantities.

Symbols standing for vectors are usually printed in boldface as **a**; this is also the convention adopted in this encyclopedia. Other conventions includes or __ a__, especially in handwriting. The

Vectors are usually shown in graphs or other diagrams as arrows, as illustrated below:

In the figure above, the arrow can also be written as or *AB*

In order to calculate with vectors, the graphical representation is too cumbersome. Vectors in a *n*-dimensional Euclidean spaces can be represented as a linear combination of *n* mutually prependicular *unit vectors*. In this article, we will consider **R**^{3} as an example. In **R**^{3}, we usually denote the unit vectors parallel to the *x*-, *y*- and *z*-axes by **i**, **j** and **k** respectively. Any vector **a** in **R**^{3} can be written as **a** = *a*_{1}**i** + *a*_{2}**j** + *a*_{3}**k** with real numbers *a*_{1}, *a*_{2} and *a*_{3} which are uniquely determined by **a**. Sometimes **a** is then also written as a 3-by-1 or 1-by-3 matrix:

the length of the vector **a** = *a*_{1}**i** + *a*_{2}**j** + *a*_{3}**k** can be computed as

Let **a**=*a*_{1}**i** + *a*_{2}**j** + *a*_{3}**k** and **b**=*b*_{1}**i** + *b*_{2}**j** + *b*_{3}**k**.

This addition method is sometimes called the *parallelogram rule* because **a** and **b** form the sides of a parallelogram and **a** + **b** is one of the diagonals. If **a** and **b** are bound vectors, then the addition is only defined if **a** and **b** have the same base point, which will then also be the base point of **a** + **b**. One can check geometrically that **a** + **b** = **b** + **a** and (**a** + **b**) + **c** = **a** + (**b** + **c**).

The difference of **a** and **b** is:

If **a** and **b** are bound vectors, then the subtraction is only defined if they share the same base point which will then also become the base point of their difference. This operations deserves the name "subtraction" because (**a** - **b**) + **b** = **a**.

A vector may also be multiplied by a real number *r*. Numbers are often called **scalars** to distinguish them from vectors, and this operation is therefore called **scalar multiplication**. The resulting vector is:

Here it is important to check that the scalar multiplication is compatible with vector addition in the following sense: *r*(**a** + **b**) = *r***a** + *r***b** for all vectors **a** and **b** and all scalars *r*. One can also show that **a** - **b** = **a** + (-1)**b**.

The set of all geometrical vectors, together with the operations of vector addition and scalar multiplication, satisfies all the axioms of a vector space. Similarly, the set of all bound vectors with a common base point forms a vector space. This is where the term "vector space" originated.

The *dot product* of two vectors **a** and **b** (also called the *inner product*, or, since its result is a scalar, the *scalar product*) is denoted by **a**·**b** or sometimes by (**a**, **b**) and is defined as:

The cross product (also *vector product* or *outer product*) differs from the dot product primarily in that the result of a cross product of two vectors is a vector.
While everything that was said above can be generalized in a straightforward manner to more than three dimensions, the cross product is only meaningful in three dimensions (although a related product exists in seven dimensions - see below).
The cross product, denoted **a**×**b**, is a vector perpendicular to both **a** and **b** and is defined as:

In such a system, **a**×**b** is defined so that **a**, **b** and **a**×**b** also becomes a right handed system. If **i**, **j**, **k** is left-handed, then **a**, **b** and **a**×**b** is defined to be left-handed. Because the cross product depends on the choice of coordinate systems, its result is referred to as a pseudovector. Fortunately, in nature cross products tend to come in pairs, so that the "handedness" of the coordinate system is undone by a second cross product.

The length of **a**×**b** can be interpreted as the area of the parallelogram having **a** and **b** as sides.

In coordinates, if the three vectors are thought of as rows, the scalar triple product is simply the determinant of the 3-by-3 matrix having the three vectors as rows. The scalar triple product is linear in all three entries and anti-symmetric in the following sense:

- Online vector identities (pdf)
- Vectors at Wikibooks