Vector notation
Encyclopedia
This page is an overview of the common notations used when working with vectors, which may be spatial or more abstract members of vector space
s.
The common typographic convention for representing a vector is upright boldface type, as in v for a vector named ‘v’. In handwriting, where boldface type is either unavailable or unwieldy, vectors are often represented with right-pointing arrows or harpoons above their names, as in
. Shorthand notations include tildes and straight lines placed above or below the name of a vector.
specified by components that define a rectangle
(or rectangular prism in three dimensions, and similar shapes in greater dimensions). The starting point and terminal point of the vector lie at opposite ends of the rectangle (or prism, etc.).
can be specified using an ordered set of components, enclosed in either parentheses or angle brackets.
In a general sense, an n-dimensional vector v can be specified in either of the following forms:
Where v1, v2, …, vn − 1, vn are the components of v.
can also be specified as a row or column matrix
containing the ordered set of components. A vector specified as a row matrix is known as a row vector; one specified as a column matrix is known as a column vector.
Again, an n-dimensional vector
can be specified in either of the following forms using matrices:
Where v1, v2, …, vn − 1, vn are the components of v. In some advanced contexts, a row and a column vector have different meaning; see covariance and contravariance of vectors.
(or fewer dimensions, such as 
where vz below is zero) can be specified as the sum of the scalar multiples of the components of the vector with the members of the standard basis
in
. The basis is represented with the unit vectors 
, 
, and 
.
A three-dimensional vector v can be specified in the following form, using engineering notation:
Where vx, vy, and vz are the magnitudes of the components of v.
(or length) and a direction (or angle). It is akin to an arrow in the polar coordinate system
. The magnitude, typically represented as r, is the length from the starting point of the vector to its endpoint. The angle, typically represented as θ (the Greek
letter theta
), is measured as the offset from the horizontal (or a line collinear with the x-axis in the positive direction). The angle is typically reduced to lie within the range
radians or 
.
.
A two-dimensional polar vector v can be represented as any of the following, using either ordered pair or matrix notation:
Where r is the magnitude, θ is the angle, and the angle symbol (
) is optional.
A two-dimensional vector whose magnitude is 5 units and whose direction is π/9 radians (20°) can be specified using either of the following forms:
. A cylindrical vector is specified by a distance in the xy-plane, an angle, and a distance from the xy-plane (a height). The first distance, usually represented as r or ρ (the Greek letter rho
), is the magnitude of the projection of the vector onto the xy-plane. The angle, usually represented as θ or φ (the Greek letter phi
), is measured as the offset from the line collinear with the x-axis in the positive direction; the angle is typically reduced to lie within the range
. The second distance, usually represented as h or z, is the distance from the xy-plane to the endpoint of the vector.
as a third component to form ordered triplets (again, a subset of ordered set notation) and matrices. The angle may be prefixed with the angle symbol (
); the distance-angle-distance combination distinguishes cylindrical vectors in this notation from spherical vectors in similar notation.
A three-dimensional cylindrical vector v can be represented as any of the following, using either ordered triplet or matrix notation:
Where r is the magnitude of the projection of v onto the xy-plane, θ is the angle between the positive x-axis and v, and h is the height from the xy-plane to the endpoint of v. Again, the angle symbol (
) is optional.
A three-dimensional vector, the magnitude of whose projection onto the xy-plane is 5 units, whose angle from the positive x-axis is π/9 radians (20°), and whose height from the xy-plane is 3 units can be specified in any of the following forms:
. A spherical vector is specified by a magnitude, an azimuth angle, and a zenith angle. The magnitude is usually represented as ρ. The azimuth angle, usually represented as θ, is the offset from the line collinear with the x-axis in the positive direction. The zenith angle, usually represented as φ, is the offset from the line collinear with the z-axis in the positive direction. Both angles are typically reduced to lie within the range from zero (inclusive) to 2π (exclusive).
); the prefix should be used consistently to produce the distance-angle-angle combination that distinguishes spherical vectors from cylindrical ones.
A three-dimensional spherical vector v can be represented as any of the following, using either ordered triplet or matrix notation:
Where ρ is the magnitude, θ is the azimuth angle, and φ is the zenith angle.
A three-dimensional vector whose magnitude is 5 units, whose azimuth angle is π/9 radians (20°), and whose zenith angle is π/4 radians (45°) can be specified as:
, the operations of vector addition and scalar multiplication are defined. Normed vector space
s also define an operation known as the norm
(or determination of magnitude). Inner product space
s also define an operation known as the inner product. In
, the inner product is known as the dot product
. In
and 
, an additional operation known as the cross product
is also defined.
is represented in the same manners as algebraic multiplication. A scalar beside a vector (either or both of which may be in parentheses) implies scalar multiplication. The two common operators, a dot and a rotated cross, are also acceptable (although the rotated cross is almost never used), but they risk confusion with dot products and cross products, which operate on two vectors. The product of a scalar c with a vector v can be represented in any of the following fashions:
Vector subtraction is performed by adding the scalar multiple of −1 with the second vector operand to the first vector operand. This can be represented by the use of the minus sign as an operator. The difference between two vectors u and v can be represented in either of the following fashions:
Scalar division is performed by multiplying the vector operand with the numeric inverse of the scalar operand. This can be represented by the use of the fraction bar or division signs as operators. The quotient of a vector v and a scalar c can be represented in any of the following forms:
of a vector is represented with double bars on both sides of the vector. The norm of a vector v can be represented as:
The norm is also sometimes represented with single bars, like
, but this can be confused with absolute value
(which is a type of norm).
(also known as the scalar product, not to be confused with scalar multiplication) of two vectors is represented as an ordered pair enclosed in angle brackets. The inner product of two vectors u and v would be represented as:
, the inner product is also known as the dot product
. In addition to the standard inner product notation, the dot product notation (using the dot as an operator) can also be used (and is more common). The dot product of two vectors u and v can be represented as:
In some older literature, the dot product is implied between two vectors written side-by-side. This notation can be confused with the dyadic product between two vectors.
of two vectors (in
) is represented using the rotated cross as an operator. The cross product of two vectors u and v would be represented as:
In some older literature, the following notation is used for the cross product between u and v:
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s.
The common typographic convention for representing a vector is upright boldface type, as in v for a vector named ‘v’. In handwriting, where boldface type is either unavailable or unwieldy, vectors are often represented with right-pointing arrows or harpoons above their names, as in
. Shorthand notations include tildes and straight lines placed above or below the name of a vector.Rectangular vectors
A rectangular vector is a coordinate vectorCoordinate vector
In linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or, equivalently, as an element of the coordinate space Fn....
specified by components that define a rectangle
Rectangle
In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...
(or rectangular prism in three dimensions, and similar shapes in greater dimensions). The starting point and terminal point of the vector lie at opposite ends of the rectangle (or prism, etc.).
Ordered set notation
A rectangular vector in can be specified using an ordered set of components, enclosed in either parentheses or angle brackets.In a general sense, an n-dimensional vector v can be specified in either of the following forms:
Where v1, v2, …, vn − 1, vn are the components of v.
Matrix notation
A rectangular vector in can also be specified as a row or column matrixMatrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
containing the ordered set of components. A vector specified as a row matrix is known as a row vector; one specified as a column matrix is known as a column vector.
Again, an n-dimensional vector
can be specified in either of the following forms using matrices:
Where v1, v2, …, vn − 1, vn are the components of v. In some advanced contexts, a row and a column vector have different meaning; see covariance and contravariance of vectors.
Engineering notation
A rectangular vector in (or fewer dimensions, such as where vz below is zero) can be specified as the sum of the scalar multiples of the components of the vector with the members of the standard basisBasis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
in
. The basis is represented with the unit vectors , , and .A three-dimensional vector v can be specified in the following form, using engineering notation:
Where vx, vy, and vz are the magnitudes of the components of v.
Polar vectors
A polar vector is a vector in two dimensions specified as a magnitudeMagnitude (mathematics)
The magnitude of an object in mathematics is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....
(or length) and a direction (or angle). It is akin to an arrow in the polar coordinate system
Polar coordinate system
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction....
. The magnitude, typically represented as r, is the length from the starting point of the vector to its endpoint. The angle, typically represented as θ (the Greek
Greek alphabet
The Greek alphabet is the script that has been used to write the Greek language since at least 730 BC . The alphabet in its classical and modern form consists of 24 letters ordered in sequence from alpha to omega...
letter theta
Theta
Theta is the eighth letter of the Greek alphabet, derived from the Phoenician letter Teth...
), is measured as the offset from the horizontal (or a line collinear with the x-axis in the positive direction). The angle is typically reduced to lie within the range
radians or .Ordered set and matrix notations
Polar vectors can be specified using either ordered pair notation (a subset of ordered set notation using only two components) or matrix notation, as with rectangular vectors. In these forms, the first component of the vector is r (instead of v1) and the second component is θ (instead of v2). To differentiate polar vectors from rectangular vectors, the angle may be prefixed with the angle symbol,.A two-dimensional polar vector v can be represented as any of the following, using either ordered pair or matrix notation:
Where r is the magnitude, θ is the angle, and the angle symbol (
) is optional.Direct notation
Polar vectors can also be specified using simplified autonomous equations that define r and θ explicitly. This can be unwieldy, but is useful for avoiding the confusion with two-dimensional rectangular vectors that arises from using ordered pair or matrix notation.A two-dimensional vector whose magnitude is 5 units and whose direction is π/9 radians (20°) can be specified using either of the following forms:
Cylindrical vectors
A cylindrical vector is an extension of the concept of polar vectors into three dimensions. It is akin to an arrow in the cylindrical coordinate systemCylindrical coordinate system
A cylindrical coordinate system is a three-dimensional coordinate systemthat specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis...
. A cylindrical vector is specified by a distance in the xy-plane, an angle, and a distance from the xy-plane (a height). The first distance, usually represented as r or ρ (the Greek letter rho
Rho
Rho is the 17th letter of the Greek alphabet. In the system of Greek numerals, it has a value of 100. It is derived from Semitic resh "head"...
), is the magnitude of the projection of the vector onto the xy-plane. The angle, usually represented as θ or φ (the Greek letter phi
Phi
Phi may refer to:In language:*Phi, the Greek letter Φ,φ, the symbol for voiceless bilabial fricativeIn mathematics:*The Golden ratio*Euler's totient function*A statistical measure of association reported with the chi-squared test...
), is measured as the offset from the line collinear with the x-axis in the positive direction; the angle is typically reduced to lie within the range
. The second distance, usually represented as h or z, is the distance from the xy-plane to the endpoint of the vector.Ordered set and matrix notations
Cylindrical vectors are specified like polar vectors, where the second distance component is concatenatedConcatenation
In computer programming, string concatenation is the operation of joining two character strings end-to-end. For example, the strings "snow" and "ball" may be concatenated to give "snowball"...
as a third component to form ordered triplets (again, a subset of ordered set notation) and matrices. The angle may be prefixed with the angle symbol (
); the distance-angle-distance combination distinguishes cylindrical vectors in this notation from spherical vectors in similar notation.A three-dimensional cylindrical vector v can be represented as any of the following, using either ordered triplet or matrix notation:
Where r is the magnitude of the projection of v onto the xy-plane, θ is the angle between the positive x-axis and v, and h is the height from the xy-plane to the endpoint of v. Again, the angle symbol (
) is optional.Direct notation
A cylindrical vector can also be specified directly, using simplified autonomous equations that define r (or ρ), θ (or φ), and h (or z). Consistency should be used when choosing the names to use for the variables; ρ should not be mixed with θ and so on.A three-dimensional vector, the magnitude of whose projection onto the xy-plane is 5 units, whose angle from the positive x-axis is π/9 radians (20°), and whose height from the xy-plane is 3 units can be specified in any of the following forms:
Spherical vectors
A spherical vector is another method for extending the concept of polar vectors into three dimensions. It is akin to an arrow in the spherical coordinate systemSpherical coordinate system
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its inclination angle measured from a fixed zenith direction, and the azimuth angle of...
. A spherical vector is specified by a magnitude, an azimuth angle, and a zenith angle. The magnitude is usually represented as ρ. The azimuth angle, usually represented as θ, is the offset from the line collinear with the x-axis in the positive direction. The zenith angle, usually represented as φ, is the offset from the line collinear with the z-axis in the positive direction. Both angles are typically reduced to lie within the range from zero (inclusive) to 2π (exclusive).
Ordered set and matrix notations
Spherical vectors are specified like polar vectors, where the zenith angle is concatenated as a third component to form ordered triplets and matrices. The azimuth and zenith angles may be both prefixed with the angle symbol (); the prefix should be used consistently to produce the distance-angle-angle combination that distinguishes spherical vectors from cylindrical ones.A three-dimensional spherical vector v can be represented as any of the following, using either ordered triplet or matrix notation:
Where ρ is the magnitude, θ is the azimuth angle, and φ is the zenith angle.
Direct notation
Like polar and cylindrical vectors, spherical vectors can be specified using simplified autonomous equations, in this case for ρ, θ, and φ.A three-dimensional vector whose magnitude is 5 units, whose azimuth angle is π/9 radians (20°), and whose zenith angle is π/4 radians (45°) can be specified as:
Operations
In any given vector spaceVector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
, the operations of vector addition and scalar multiplication are defined. Normed vector space
Normed vector space
In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....
s also define an operation known as the norm
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
(or determination of magnitude). Inner product space
Inner product space
In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...
s also define an operation known as the inner product. In
, the inner product is known as the dot productDot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
. In
and , an additional operation known as the cross productCross product
In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...
is also defined.
Vector addition
Vector addition is represented with the plus sign used as an operator between two vectors. The sum of two vectors u and v would be represented as:Scalar multiplication
Scalar multiplicationScalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra . In an intuitive geometrical context, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction...
is represented in the same manners as algebraic multiplication. A scalar beside a vector (either or both of which may be in parentheses) implies scalar multiplication. The two common operators, a dot and a rotated cross, are also acceptable (although the rotated cross is almost never used), but they risk confusion with dot products and cross products, which operate on two vectors. The product of a scalar c with a vector v can be represented in any of the following fashions:
Vector subtraction and scalar division
Using the algebraic properties of subtraction and division, along with scalar multiplication, it is also possible to “subtract” two vectors and “divide” a vector by a scalar.Vector subtraction is performed by adding the scalar multiple of −1 with the second vector operand to the first vector operand. This can be represented by the use of the minus sign as an operator. The difference between two vectors u and v can be represented in either of the following fashions:
Scalar division is performed by multiplying the vector operand with the numeric inverse of the scalar operand. This can be represented by the use of the fraction bar or division signs as operators. The quotient of a vector v and a scalar c can be represented in any of the following forms:
Norm
The normNorm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
of a vector is represented with double bars on both sides of the vector. The norm of a vector v can be represented as:
The norm is also sometimes represented with single bars, like
, but this can be confused with absolute valueAbsolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
(which is a type of norm).
Inner product
The inner productInner product space
In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...
(also known as the scalar product, not to be confused with scalar multiplication) of two vectors is represented as an ordered pair enclosed in angle brackets. The inner product of two vectors u and v would be represented as:
Dot product
In, the inner product is also known as the dot productDot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
. In addition to the standard inner product notation, the dot product notation (using the dot as an operator) can also be used (and is more common). The dot product of two vectors u and v can be represented as:
In some older literature, the dot product is implied between two vectors written side-by-side. This notation can be confused with the dyadic product between two vectors.
Cross product
The cross productCross product
In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...
of two vectors (in
) is represented using the rotated cross as an operator. The cross product of two vectors u and v would be represented as:In some older literature, the following notation is used for the cross product between u and v: