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 |

## Definitions

Informally, a**vector**is a quantity, characterized by a number (indicating size or "magnitude") and a direction, that is often represented graphically by an arrow. Examples are "moving north at 90 m.p.h" or "pulling towards the center of Earth with a force of 70 Newtons".

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.

#### Generalizations

In mathematics, a**vector**over a field k is any element of a vector space. The spatial vectors of this article are a very special case of this general definition (they are

*not*simply any element of

**R**

^{d}in

*d*dimensions), which includes a variety of mathematical objects (algebras, the set of all functions from a given domain to a given linear range, and linear transformations). Note that under this definition, a tensor is a special vector!

## Representation of a vector

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

*length*or

*magnitude*or

*norm*of the vector

**a**is denoted by |

**a**|.

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

*A*is called the

*tail*,

*base*,

*start*, or

*origin*; point

*B*is called the

*head*,

*tip*,

*endpoint*, or

*destination*. The length of the arrow represents the vector's magnitude, while the direction in which the arrow points represents the vector's direction.

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:

*a*

_{1},

*a*

_{2}and

*a*

_{3}on the specific choice of coordinate system

**i**,

**j**and

**k**.

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

## Vector Equality

## Vector Addition and Subtraction

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

**b**at the tip of the arrow

**a**, and then drawing an arrow from the start of

**a**to the tip of

**b**. The new arrow drawn represents the vector

**a**+

**b**, as illustrated below:

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:

**b**from

**a**, place the ends of

**a**and

**b**at the same point, and then draw an arrow from the tip of

**b**to the tip of

**a**. That arrow represents the vector

**a**-

**b**, as illustrated below:

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:

*r*

**a**is |

*r*||

**a**|. If the scalar is negative, it also changes the direction of the vector by 180

^{o}. Two examples (

*r*= -1 and

*r*= 2) are given below:

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.

## Dot Product

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:

**a**and

**b**(see trigonometric function for an explanation of cosine). Geometrically, this means that

**a**and

**b**are drawn with a common start point and then the length of

**a**is multiplied with the length of that component of

**b**that points in the same direction as

**a**. This operation is often useful in physics; for instance, work is the dot product of force and displacement.

## Cross Product

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:

**a**and

**b**, and

**n**is a unit vector perpendicular to both

**a**and

**b**. The problem with this definition is that there are

*two*unit vectors perpendicular to both

**b**and

**a**. Which vector is the correct one depends upon the

*orientation*of the vector space, i.e. on the

*handedness*of the coordinate system. The coordinate system

**i**,

**j**,

**k**is called

*right handed*, if the three vectors are situated like the thumb, index finger and middle finger (pointing straight up from your palm) of your right hand. Graphically the cross product can be represented by

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.

## Scalar Triple Product

The*scalar triple product*(also called the

*box product*or

*mixed triple product*) isn't really a new operator, but a way of applying the other two multiplication operators to three vectors. The scalar triple product is denoted by (

**a**

**b**

**c**) and defined as:

**a**,

**b**and

**c**are oriented like the coordinate system

**i**,

**j**and

**k**.

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:

*pseudoscalar*: under a coordinate inversion (

**x**goes to -

**x**), it flips sign.

## External links

- Online vector identities (pdf)
- Vectors at Wikibooks