 # Vector algebra

## Vector algebra Topics

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### Vector triple product

The vector triple product identity is also known as the BAC-CABidentity, and can be written in the form(1)(2)(3)

### Scalar multiplication

Scalar multiplication refers to the multiplication of a vector by a constant , producing a vector in the same (for ) or opposite (for ) direction but of different length. Scalar multiplication is indicated in the Wolfram Language by placing a scalar next to a vector (with or without an optional asterisk), sa1, a2, ..., an.

### Green's identities

Green's identities are a set of three vector derivative/integral identities which can be derived starting with the vector derivative identities(1)and(2)where is the divergence, is the gradient, is the Laplacian, and is the dot product. From the divergence theorem,(3)Plugging (2) into (3),(4)This is Green's first identity.Subtracting (2) from (1),(5)Therefore,(6)This is Green's second identity.Let have continuous first partial derivatives and be harmonic inside the region of integration. Then Green's third identity is(7)(Kaplan 1991, p. 361).

### Vector transformation law

The set of quantities are components of an -dimensional vector iff, under rotation,(1)for , 2, ..., . The direction cosines between and are(2)They satisfy the orthogonality condition(3)where is the Kronecker delta.

where is the gradient, and the integral is a line integral. It is this relationship which makes the definition of a scalar potential function so useful in gravitation and electromagnetism as a concise way to encode information about a vector field.

### Vector sum

A vector sum is the result of adding two or more vectors together via vector addition. It is denoted using the normal plus sign, i.e., the vector sum of vectors , , and is written .

### Reynolds transport theorem

The Reynolds transport theorem, also called simply the Reynolds theorem, is an important result in fluid mechanics that's often considered a three-dimensional analog of the Leibniz integral rule. Given any scalar quantity associated with a moving fluid, the general form of Reynolds transport theorem saysHere, is the convective derivative, is the usual gradient, denotes the material volume at time , and denotes the velocity vector.Because of its relation to the Leibniz rule, the Reynolds transport theorem is sometimes called the Leibniz-Reynolds transport theorem.Worth noting is the large number of variants of Reynolds transport theorem present in the literature. Indeed, the formula is extremely general and can be applied to a variety of contexts in vastly many circumstances. As such, different literature will inevitably have equations which often look different than the above equation in both appearance and complexity...

The term "gradient" has several meanings in mathematics. The simplest isas a synonym for slope.The more general gradient, called simply "the" gradient in vector analysis, is a vector operator denoted and sometimes also called del or nabla. It is most often applied to a real function of three variables , and may be denoted(1)For general curvilinear coordinates, thegradient is given by(2)which simplifies to(3)in Cartesian coordinates.The direction of is the orientation in which the directional derivative has the largest value and is the value of that directional derivative. Furthermore, if , then the gradient is perpendicular to the level curve through if and perpendicular to the level surface through if .In tensor notation, let(4)be the line element in principal form. Then(5)For a matrix ,(6)For expressions giving the gradient in particular coordinate systems, see curvilinearcoordinates...

### Vector spherical harmonic

The spherical harmonics can be generalized to vector spherical harmonics by looking for a scalar function and a constant vector such that(1)(2)(3)(4)so(5)Now interchange the order of differentiation and use the fact that multiplicative constants can be moving inside and outside the derivatives to obtain(6)(7)(8)and(9)(10)Putting these together gives(11)so satisfies the vector Helmholtz differential equation if satisfies the scalar Helmholtz differential equation(12)Construct another vector function(13)which also satisfies the vector Helmholtzdifferential equation since(14)(15)(16)(17)(18)which gives(19)We have the additional identity(20)(21)(22)(23)(24)In this formalism, is called the generating function and is called the pilot vector. The choice of generating function is determined by the symmetry of the scalar equation, i.e., it is chosen to solve the desired scalar differential equation. If is taken as(25)where..

### Vector space span

The span of subspace generated by vectors and isA set of vectors can be tested to see if they span -dimensional space using the following Wolfram Language function: SpanningVectorsQ[m_List?MatrixQ] := (NullSpace[m] == {})

### Ray

There are several definitions of a ray.When viewed as a vector, a ray is a vector from a point to a point .In geometry, a ray is usually taken as a half-infinite line (also known as a half-line) with one of the two points and taken to be at infinity.

### Vector space

A vector space is a set that is closed under finite vector addition and scalar multiplication. The basic example is -dimensional Euclidean space , where every element is represented by a list of real numbers, scalars are real numbers, addition is componentwise, and scalar multiplication is multiplication on each term separately.For a general vector space, the scalars are members of a field , in which case is called a vector space over .Euclidean -space is called a real vector space, and is called a complex vector space.In order for to be a vector space, the following conditions must hold for all elements and any scalars : 1. Commutativity:(1)2. Associativity of vectoraddition:(2)3. Additive identity: For all ,(3)4. Existence of additive inverse: For any , there exists a such that(4)5. Associativity of scalar multiplication:(5)6. Distributivity of scalar sums:(6)7. Distributivity of vector sums:(7)8. Scalar multiplication identity:(8)Let..

### Dual basis

Given a contravariant basis , its dual covariant basis is given bywhere is the metric and is the mixed Kronecker delta. In Euclidean space with an orthonormal basis,so the basis and its dual are the same.

There are a number of algebraic identities involving sets of four vectors. An identity known as Lagrange's identity is given by(1)(Bronshtein and Semendyayev 2004, p. 185).Letting , a number of other useful identities include(2)(3)(4)(5)(6)where denotes the scalar triple product. Equation (◇) turns out to be relevant in the computation of the point-line distance in three dimensions.

The vector from the origin to the current position. It is also called the position vector. The derivative of satisfieswhere is the magnitude of the velocity (i.e., the speed).

### Dot product

The dot product can be defined for two vectors and by(1)where is the angle between the vectors and is the norm. It follows immediately that if is perpendicular to . The dot product therefore has the geometric interpretation as the length of the projection of onto the unit vector when the two vectors are placed so that their tails coincide.By writing(2)(3)it follows that (1) yields(4)(5)(6)(7)So, in general,(8)(9)This can be written very succinctly using Einsteinsummation notation as(10)The dot product is implemented in the Wolfram Language as Dot[a, b], or simply by using a period, a . b.The dot product is commutative(11)and distributive(12)The associative property is meaningless for the dot product because is not defined since is a scalar and therefore cannot itself be dotted. However, it does satisfy the property(13)for a scalar.The derivative of a dot product of vectorsis(14)The dot product is invariant under rotations(15)(16)(17)(18)(19)(20)where..

### Pseudovector

A typical vector (i.e., a vector such as the radius vector ) is transformed to its negative under inversion of its coordinate axes. Such "proper" vectors are known as polar vectors. A vector-like object which is invariant under inversion is called a pseudovector, also called an axial vector (as a result of such vectors frequently arising as vectors describing rotation; Arfken 1985, p. 128; Morse and Feshbach 1953). The cross product(1)is a pseudovector, whereas the vector tripleproduct(2)is a polar vector. (Polar) vectors and pseudovectors are interrelated in the following ways under application of the cross product,(3)(4)Examples of pseudovectors therefore include the angular velocity vector , angular momentum , torque , auxiliary magnetic field , and magnetic dipole moment .Given a transformation matrix ,(5)where Einstein summation has been used...

### Divergenceless field

A divergenceless vector field, also called a solenoidal field, is a vector field for which . Therefore, there exists a such that . Furthermore, can be written as(1)(2)where(3)(4)(5)(6)Following Lamb's 1932 treatise (Lamb 1993), and are called toroidal field and poloidal field.

### Vector orientation

Let be the angle between two vectors. If , the vectors are positively oriented. If , the vectors are negatively oriented.Two vectors in the planeare positively oriented iff the determinantand are negatively oriented iff the determinant .

### Divergence theorem

The divergence theorem, more commonly known especially in older literature as Gauss's theorem (e.g., Arfken 1985) and also known as the Gauss-Ostrogradsky theorem, is a theorem in vector calculus that can be stated as follows. Let be a region in space with boundary . Then the volume integral of the divergence of over and the surface integral of over the boundary of are related by(1)The divergence theorem is a mathematical statement of the physical fact that, in the absence of the creation or destruction of matter, the density within a region of space can change only by having it flow into or away from the region through its boundary.A special case of the divergence theorem follows by specializing to the plane. Letting be a region in the plane with boundary , equation (1) then collapses to(2)If the vector field satisfies certain constraints, simplified forms can be used. For example, if where is a constant vector , then(3)But(4)so(5)(6)and(7)But..

### Vector ordering

If the first nonzero component of the vector difference is , then . If the first nonzero component of is , then .

### Polar vector

There are two different definitions of "polar vector."In elementary math, the term "polar vector" is used to refer to a representation of a vector as a vector magnitude (length) and angle, which is equivalent to specifying its endpoints in polar coordinates (illustrated above).In physics, a polar vector is a vector such as the radius vector that reverses sign when the coordinate axes are reversed. Polar vectors are the type of vector usually simply known as "vectors." In contrast, pseudovectors (also called axial vectors) do not reverse sign when the coordinate axes are reversed. Examples of polar vectors include , the velocity vector , momentum , and force . The cross product of two polar vectors is a pseudovector.

### Divergence

The divergence of a vector field , denoted or (the notation used in this work), is defined by a limit of the surface integral(1)where the surface integral gives the value of integrated over a closed infinitesimal boundary surface surrounding a volume element , which is taken to size zero using a limiting process. The divergence of a vector field is therefore a scalar field. If , then the field is said to be a divergenceless field. The symbol is variously known as "nabla" or "del."The physical significance of the divergence of a vector field is the rate at which "density" exits a given region of space. The definition of the divergence therefore follows naturally by noting that, in the absence of the creation or destruction of matter, the density within a region of space can change only by having it flow into or out of the region. By measuring the net flux of content passing through a surface surrounding the region of space,..

### Vector norm

Given an -dimensional vector(1)a general vector norm , sometimes written with a double bar as , is a nonnegative norm defined such that 1. when and iff . 2. for any scalar . 3. . In this work, a single bar is used to denote a vector norm, absolute value, or complex modulus, while a double bar is reserved for denoting a matrix norm.The vector norm for , 2, ... is defined as(2)The -norm of vector is implemented as Norm[v, p], with the 2-norm being returned by Norm[v].The special case is defined as(3)The most commonly encountered vector norm (often simply called "the norm" of a vector, or sometimes the magnitude of a vector) is the L2-norm, given by(4)This and other types of vector norms are summarized in the following table, together with the value of the norm for the example vector .namesymbolvalueapprox.-norm66.000-norm3.742-norm3.302-norm3.146-norm33.000..

### Perpendicular vector

A vector perpendicular to a given vector is a vector (voiced "-perp") such that and form a right angle.In the plane, there are two vectors perpendicular to any given vector, one rotated counterclockwise and the other rotated clockwise. Hill (1994) defines to be the perpendicular vector obtained from an initial vector(1)by a counterclockwise rotation by , i.e.,(2)In the plane, a vector perpendicular to can therefore be obtained by transposing the Cartesian components and taking the minus sign of one. This operation is implemented in the Wolfram Language as Cross[ax, ay].In three dimensions, there are an infinite number of vectors perpendicular to a given vector, all satisfying the equations(3)

### Perpendicular

Two lines, vectors, planes, etc., are said to be perpendicular if they meet at a right angle. In , two vectors and are perpendicular if their dot product(1)In , a line with slope is perpendicular to a line with slope . Perpendicular objects are sometimes said to be "orthogonal."In the above figure, the line segment is perpendicular to the line segment . This relationship is commonly denoted with a small square at the vertex where perpendicular objects meet, as shown above, and is denoted .Two trilinear lines(2)(3)are perpendicular if(4)(Kimberling 1998, p. 29).

### Direction

The direction from an object to another object can be specified as a vector with tail at and head at . However, since this vector has length equal to the distance between the objects in addition to encoding the direction from the first to the second, it is natural to instead consider the unit vector (sometimes called the direction vector), which decouples the distance from the direction.

### Vector multiplication

Although the multiplication of one vector by another is not uniquely defined (cf. scalar multiplication, which is multiplication of a vector by a scalar), several types of useful vector products can be defined, as summarized in the following table.product namesymbolresultdot productscalarcross productpseudovectorperp dot productscalarvector direct producttensorVector multiplication can also be defined for vectors taken three at a time, as summarized in the following table.product namesymbolresultvector triple productvectorscalar triple productpseudoscalarA number of vector quadruple productscan also be defined.

### Perp dot product

The "perp dot product" for and vectors in the plane is a modification of the two-dimensional dot product in which is replaced by the perpendicular vector rotated to the left defined by Hill (1994). It satisfies the identities(1)(2)where is the angle from vector to vector .

### Vector laplacian

A vector Laplacian can be defined for a vector by(1)where the notation is sometimes used to distinguish the vector Laplacian from the scalar Laplacian (Moon and Spencer 1988, p. 3). In tensor notation, is written , and the identity becomes(2)(3)(4)A tensor Laplacian may be similarly defined.In cylindrical coordinates, the vectorLaplacian is given by(5)In spherical coordinates, the vector Laplacianis(6)

### Parallelogram law

The parallelogram law gives the rule for vector addition of vectors and . The sum of the vectors is obtained by placing them head to tail and drawing the vector from the free tail to the free head.Let denote the norm of a quantity. Then the quantities and are said to satisfy the parallelogram law ifIf the norm is defined as (the so-called L2-norm), then the law will always hold.

### Darboux vector

The rotation vector of the trihedron of a curve with curvature when a point moves along a curve with unit speed. It is given by(1)where is the torsion, the tangent vector, and the binormal vector. The Darboux vector field satisfies(2)(3)(4)

### Vector integral

The following vector integrals are related to the curltheorem. If(1)then(2)If(3)then(4)The following are related to the divergence theorem.If(5)then(6)Finally, if(7)then(8)

### Curl theorem

A special case of Stokes' theorem in which is a vector field and is an oriented, compact embedded 2-manifold with boundary in , and a generalization of Green's theorem from the plane into three-dimensional space. The curl theorem states(1)where the left side is a surface integral andthe right side is a line integral.There are also alternate forms of the theorem. If(2)then(3)and if(4)then(5)

### Curl

The curl of a vector field, denoted or (the notation used in this work), is defined as the vector field having magnitude equal to the maximum "circulation" at each point and to be oriented perpendicularly to this plane of circulation for each point. More precisely, the magnitude of is the limiting value of circulation per unit area. Written explicitly,(1)where the right side is a line integral around an infinitesimal region of area that is allowed to shrink to zero via a limiting process and is the unit normal vector to this region. If , then the field is said to be an irrotational field. The symbol is variously known as "nabla" or "del."The physical significance of the curl of a vector field is the amount of "rotation" or angular momentum of the contents of given region of space. It arises in fluid mechanics and elasticity theory. It is also fundamental in the theory of electromagnetism, where it arises in..

### Cross product

For vectors and in , the cross product in is defined by(1)(2)where is a right-handed, i.e., positively oriented, orthonormal basis. This can be written in a shorthand notation that takes the form of a determinant(3)where , , and are unit vectors. Here, is always perpendicular to both and , with the orientation determined by the right-hand rule.Special cases involving the unit vectors in three-dimensionalCartesian coordinates are given by(4)(5)(6)The cross product satisfies the general identity(7)Note that is not a usual polar vector, but has slightly different transformation properties and is therefore a so-called pseudovector (Arfken 1985, pp. 22-23). Jeffreys and Jeffreys (1988) use the notation to denote the cross product.The cross product is implemented in the Wolfram Language as Cross[a, b].A mathematical joke asks, "What do you get when you cross a mountain-climber with a mosquito?" The answer is, "Nothing:..

### Vector field

A vector field is a map that assigns each a vector . Several vector fields are illustrated above. A vector field is uniquely specified by giving its divergence and curl within a region and its normal component over the boundary, a result known as Helmholtz's theorem (Arfken 1985, p. 79).Vector fields can be plotted in the Wolfram Language using VectorPlot[f, x, xmin, xmax, y, ymin, ymax].Flows are generated by vector fields and vice versa. A vector field is a tangent bundle section of its tangent bundle.

### Vector division

In general, there is no unique matrix solution to the matrix equationEven in the case of parallel to , there are still multiple matrices that perform this transformation. For example, given , all the following matrices satisfy the above equation:Therefore, vector division cannot be uniquely defined in terms of matrices.However, if the vectors are represented by complex numbers or quaternions, vector division can be uniquely defined using the usual rules of complex division and quaternion algebra, respectively.

### Orthogonal basis

An orthogonal basis of vectors is a set of vectors that satisfyandwhere , are constants (not necessarily equal to 1), is the Kronecker delta, and Einstein summation has been used.If the constants are all equal to 1, then the set of vectors is called an orthonormalbasis.

### Vector direct product

Given vectors and , the vector direct product, also known as a dyadic, iswhere is the Kronecker product and is the matrix transpose. For the direct product of two 3-vectors,Note that if , then , where is the Kronecker delta.

### Convex combination

A subset of a vector space is said to be convex if for all vectors , and all scalars . Via induction, this can be seen to be equivalent to the requirement that for all vectors , and for all scalars such that . With the above restrictions on the , an expression of the form is said to be a convex combination of the vectors .The set of all convex combinations of vectors in constitute the convex hull of so, for example, if are two different vectors in the vector space , then the set of all convex combinations of and constitute the line segment between and .

### Vector difference

A vector difference is the result of subtracting one vector from another. A vector difference is denoted using the normal minus sign, i.e., the vector difference of vectors and is written .A vector difference is equivalent to a vector sum withthe orientation of the second vector reversed, i.e.,

### Convective operator

Defined for a vector field by , where is the gradient operator.Applied in arbitrary orthogonal three-dimensional coordinates to a vector field , the convective operator becomes(1)where the s are related to the metric tensors by . In Cartesian coordinates,(2)In cylindrical coordinates,(3)In spherical coordinates,(4)

### Null vector

There are several meanings of "null vector" in mathematics.1. The most common meaning of null vector is the -dimensional vector of length 0. i.e., the vector with components, each of which is 0 (Jeffreys and Jeffreys 1988, p. 64). 2. When applied to a matrix , a null vector is a nonzero vector with the property that . 3. When applied to a vector space with an associated quadratic form , a null vector is a nonzero element of for which . 4. When applied to a geometric product satisfying the contraction rule for an element of an -vector space, a null vector is a value of such that but (Sommer 2001, pp. 5-6). 5. When applied to a vector, a null vector is a nonzero vector such that for a given vector , the dot product satisfies . (This use may be slightly nonstandard, but appears for example in the Wolfram Language's FindIntegerNullVector function.) ..

### Convective derivative

The convective derivative is a derivative taken with respect to a moving coordinate system. It is also called the advective derivative, derivative following the motion, hydrodynamic derivative, Lagrangian derivative, material derivative, particle derivative, substantial derivative, substantive derivative (Tritton 1989), Stokes derivative (Kaplan 1991, pp. 189-191), or total derivative. It is given bywhere is the gradient operator and is the velocity of the fluid. This type of derivative is especially useful in the study of fluid mechanics. When applied to ,

### Convective acceleration

The acceleration of an element of fluid, given by the convective derivative of the velocity ,where is the gradient operator.

Vector addition is the operation of adding two or more vectors together into a vector sum.The so-called parallelogram law gives the rule for vector addition of two or more vectors. For two vectors and , the vector sum is obtained by placing them head to tail and drawing the vector from the free tail to the free head. In Cartesian coordinates, vector addition can be performed simply by adding the corresponding components of the vectors, so if and ,Vector addition is indicated in the Wolfram Language using a plus sign, e.g., a1, a2, ..., an+b1, b2, ..., bn.

### Normalized vector

The normalized vector of is a vector in the same direction but with norm (length) 1. It is denoted and given bywhere is the norm of . It is also called a unit vector.

### Vector

A vector is formally defined as an element of a vector space. In the commonly encountered vector space (i.e., Euclidean n-space), a vector is given by coordinates and can be specified as . Vectors are sometimes referred to by the number of coordinates they have, so a 2-dimensional vector is often called a two-vector, an -dimensional vector is often called an n-vector, and so on.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, cross product, and tensor direct product can be defined for pairs of vectors.A vector from a point to a point is denoted , and a vector may be denoted , or more commonly, . The point is often called the "tail" of the vector, and is called the vector's "head." A vector with unit length is called a unit vector..

### Normal vector

The normal vector, often simply called the "normal," to a surface is a vector which is perpendicular to the surface at a given point. When normals are considered on closed surfaces, the inward-pointing normal (pointing towards the interior of the surface) and outward-pointing normal are usually distinguished.The unit vector obtained by normalizing the normal vector (i.e., dividing a nonzero normal vector by its vector norm) is the unit normal vector, often known simply as the "unit normal." Care should be taken to not confuse the terms "vector norm" (length of vector), "normal vector" (perpendicular vector) and "normalized vector" (unit-length vector).The normal vector is commonly denoted or , with a hat sometimes (but not always) added (i.e., and ) to explicitly indicate a unit normal vector.The normal vector at a point on a surface is given by(1)where and are partial derivatives.A..

### Conservative field

The following conditions are equivalent for a conservative vector field on a particular domain : 1. For any oriented simple closed curve , the line integral . 2. For any two oriented simple curves and with the same endpoints, . 3. There exists a scalar potential function such that , where is the gradient. 4. If is simply connected, then curl . The domain is commonly assumed to be the entire two-dimensional plane or three-dimensional space. However, there are examples of fields that are conservative in two finite domains and but are not conservative in their union .Note that conditions 1, 2, and 3 are equivalent for any vector field defined in any open set , with the understanding that the curves , , and are contained in and that holds at every point of .In general, condition 4 is not equivalent to conditions 1, 2 and 3 (and counterexamples are known in which 4 does not imply the others and vice versa), although if the first derivatives of the components of are..

### Unit vector

A unit vector is a vector of length 1, sometimes also called a direction vector (Jeffreys and Jeffreys 1988). The unit vector having the same direction as a given (nonzero) vector is defined bywhere denotes the norm of , is the unit vector in the same direction as the (finite) vector . A unit vector in the direction is given bywhere is the radius vector.When considered as the th basis vector of a vector space, a unit vector may be written (or ).

### Binormal vector

(1)(2)where the unit tangent vector and unit "principal" normal vector are defined by(3)(4)Here, is the radius vector, is the arc length, is the torsion, and is the curvature. The binormal vector satisfies the remarkable identity(5)In the field of computer graphics, two orthogonal vectors tangent to a surface are frequently referred to as tangent and binormal vectors. However, for a surface, the two vectors are more properly called tangent and bitangent vectors.

### Toroidal field

A divergenceless field can be partitioned into a toroidal and a poloidal part. This separation is important in geo- and heliophysics, and in particular in dynamo theory and helioseismology. The toroidal field is defined aswhich can additionally be multiplied by a radial weighting function .This is equivalent to the definitionwhere is a scalar function and the gradient is taken in spherical coordinates.

### Biharmonic operator

The biharmonic operator, also known as the bilaplacian, is the differential operator defined bywhere is the Laplacian.In -dimensional space,

### Linearly dependent vectors

vectors , , ..., are linearly dependent iff there exist scalars , , ..., , not all zero, such that(1)If no such scalars exist, then the vectors are said to be linearly independent. In order to satisfy the criterion for linear dependence,(2)(3)In order for this matrix equation to have a nontrivial solution, the determinant must be 0, so the vectors are linearly dependent if(4)and linearly independent otherwise.Let and be -dimensional vectors. Then the following three conditions are equivalent (Gray 1997). 1. and are linearly dependent. 2. . 3. The matrix has rank less than two.

### Biharmonic equation

The differential equation obtained by applying the biharmonicoperator and setting to zero:(1)In Cartesian coordinates, the biharmonicequation is(2)(3)(4)(5)In polar coordinates (Kaplan 1984, p. 148)(6)For a radial function , the biharmonic equation becomes(7)(8)The solution to the homogeneous equation is(9)The homogeneous biharmonic equation can be separated and solved in two-dimensionalbipolar coordinates.The solution to the inhomogeneous equation(10)is given by(11)

### Jerk

The jerk is defined as the time derivative of the vector acceleration ,

### Solenoidal field

A solenoidal vector field satisfies(1)for every vector , where is the divergence. If this condition is satisfied, there exists a vector , known as the vector potential, such that(2)where is the curl. This follows from the vector identity(3)If is an irrotational field, then(4)is solenoidal. If and are irrotational, then(5)is solenoidal. The quantity(6)where is the gradient, is always solenoidal. For a function satisfying Laplace's equation(7)it follows that is solenoidal (and also irrotational).

### Abstract vector space

An abstract vector space of dimension over a field is the set of all formal expressions(1)where is a given set of objects (called a basis) and is any -tuple of elements of . Two such expressions can be added together by summing their coefficients,(2)This addition is a commutative group operation, since the zero element is and the inverse of is . Moreover, there is a natural way to define the product of any element by an arbitrary element (a so-called scalar) of ,(3)Note that multiplication by 1 leaves the element unchanged.This structure is a formal generalization of the usual vector space over , for which the field of scalars is the real field and a basis is given by . As in this special case, in any abstract vector space , the multiplication by scalars fulfils the following two distributive laws: 1. For all and all , . 2. For all and all , . These are the basic properties of the integer multiples in any commutative additive group. This special behavior of a product..

The vectors , ..., in a three-space form a normalized eutactic star iff for all in the three-space.

### Velocity vector

The idea of a velocity vector comes from classical physics. By representing the position and motion of a single particle using vectors, the equations for motion are simpler and more intuitive. Suppose the position of a particle at time is given by the position vector . Then the velocity vector is the derivative of the position,For example, suppose a particle is confined to the plane and its position is given by . Then it travels along the unit circle at constant speed. Its velocity vector is . In a diagram, it makes sense to translate the velocity vector so it originates at . In particular, it is drawn as an arrow from to .Another example is a particle traveling along a hyperbola specified parametrically by . Its velocity vector is then given by , illustrated above.Travel down the same path, but using a different function is called a reparameterization, and the chain rule describes the change in velocity. For example, the hyperbola can also be parametrized..

### Green's theorem

Green's theorem is a vector identity which is equivalent to the curl theorem in the plane. Over a region in the plane with boundary , Green's theorem states(1)where the left side is a line integral and the right side is a surface integral. This can also be written compactly in vector form as(2)If the region is on the left when traveling around , then area of can be computed using the elegant formula(3)giving a surprising connection between the area of a region and the line integral around its boundary. For a plane curve specified parametrically as for , equation (3) becomes(4)which gives the signed area enclosed by the curve.The symmetric for above corresponds to Green's theorem with and , leading to(5)(6)(7)(8)(9)However, we are also free to choose other values of and , including and , giving the "simpler" form(10)and and , giving(11)A similar procedure can be applied to compute the moment about the -axis using and as(12)and about the..

### Seifert conjecture

Every smooth nonzero vector field on the 3-sphere has at least one closed orbit. The conjecture was proposed in 1950 and proved true for Hopf maps. The conjecture was subsequently demonstrated to be false over (Schweitzer 1974), over (Harrison 1988), and finally false in general (Kuperberg 1994).

### Helmholtz's theorem

Any vector field satisfying(1)(2)may be written as the sum of an irrotationalpart and a solenoidal part,(3)where(4)(5)

### Scalar triple product

The scalar triple product of three vectors , , and is denoted and defined by(1)(2)(3)(4)(5)where denotes a dot product, denotes a cross product, denotes a determinant, and , , and are components of the vectors , , and , respectively. The scalar triple product is a pseudoscalar (i.e., it reverses sign under inversion). The scalar triple product can also be written in terms of the permutation symbol as(6)where Einstein summation has been used to sumover repeated indices.Additional identities involving the scalar triple product are(7)(8)(9)The volume of a parallelepiped whose sides are given by the vectors , , and is given by the absolute value of the scalar triple product(10)

### Hamel basis

A basis for the real numbers , considered as a vector space over the rationals , i.e., a set of real numbers such that every real number has a unique representation of the formwhere is rational and depends on .The axiom of choice is equivalent to the statement: "Every vector space has a vector space basis," and this is the only justification for the existence of a Hamel basis.

### Spinor field

In particle physics, a spinor field of order describes a particle of spin , where is an integer or half-integer. Therefore, a spinor of order contains as much information as a tensor of order . As a result of this, particles of integer spin (bosons) can be described equally well by tensor fields or spinor fields, whereas particles of half-integer spin (fermions) can be described only by spinor fields. Spinor fields describing particles of zero rest mass satisfy the zero rest mass equation.