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Encyclopedia > Vector (spatial)
A vector going from A to B.
A vector going from A to B.

A spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. A vector can be thought of as an arrow in Euclidean space, drawn from an initial point A pointing to a terminal point B. This vector is commonly denoted by In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... Look up vector in Wiktionary, the free dictionary. ... Image File history File links Vector_AB_from_A_to_B.svg‎ (All user names refer to en. ... Image File history File links Vector_AB_from_A_to_B.svg‎ (All user names refer to en. ... The magnitude of a mathematical object is its size: a property by which it can be larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

overrightarrow{AB},

indicating that the arrow points from A to B. In this way, the arrow holds all the information of the vector quantity — the magnitude is represented by the arrow's length and the direction by the direction of the arrow's head and body. This magnitude and direction are those necessary to carry one from A to B. [1]


Vectors have a variety of algebraic properties. Vectors may be scaled by stretching them out, or compressing them. They can be flipped around so as to point in the opposite direction. Two vectors sharing the same initial point can also be added or subtracted. This article is about the branch of mathematics. ... In mathematics, a homothety (or homothecy) is a transformation of space which dilates distances with respect to a fixed point called the origin. ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ... 5 - 2 = 3 (verbally, five minus two equals three) An example problem Subtraction is one of the four basic arithmetic operations; it is the inverse of addition. ...

Contents

Overview

Informally, a vector is a quantity characterized by a magnitude (in mathematics a number, in physics a number times a unit) and a direction, often represented graphically by an arrow. Sometimes, one speaks of bound or fixed vectors, which are vectors whose initial point is the origin. This is in contrast to free vectors, which are vectors whose initial point is not necessarily the origin. The magnitude of a mathematical object is its size: a property by which it can be larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs. ...


Use in physics and engineering

Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has both a magnitude and direction, such as velocity, the magnitude of which is speed. For example, the velocity "5 tfrac{meters}{second} up" could be represented by the vector (0,5). Another quantity represented by a vector is force, since it has a magnitude and direction. Vectors also describe many other physical quantities, such as displacement, acceleration, electric and magnetic fields, momentum, and angular momentum. This article is about velocity in physics. ... This article does not cite any references or sources. ... In physics, a net force acting on a body causes that body to accelerate; that is, to change its velocity. ... In Newtonian mechanics, displacement is the vector that specifies the position of a point or a particle in reference to an origin or to a previous position. ... Acceleration is the time rate of change of velocity and/or direction, and at any point on a velocity-time graph, it is given by the slope of the tangent to the curve at that point. ... In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. ... The name Magnetic Fields has been used by: A 1981 album by Jean Michel Jarre; see Magnetic Fields (album) (Les Chants Magnetiques) A computer game developer; see Magnetic Fields (computer game developer) The Magnetic Fields, a band led by Stephin Merritt For magnetic fields in general, see magnetic field. ... This article is about momentum in physics. ... This gyroscope remains upright while spinning due to its angular momentum. ...


Vectors in Cartesian space

In Cartesian coordinates, a vector can be represented by identifying the coordinates of its initial and terminal point. For instance, the points A = (1,0,0) and B = (0,1,0) in space determine the free vector pointing from the point x=1 on the x-axis to the point y=1 on the y-axis. Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...


Typically in Cartesian coordinates, one considers primarily bound vectors. A bound vector is determined by the coordinates of the terminal point, its initial point always having the coordinates of the origin O = (0,0,0). Thus the bound vector represented by (1,0,0) is a vector of unit length pointing from the origin up the positive x-axis.


The coordinate representation of vectors allows the algebraic features of vectors to be expressed in a convenient numerical fashion. For example, the sum of the vectors (1,2,3) and (-2,0,4) is the vector

(1,, 2,, 3) + (-2,, 0,, 4)=(1-2,, 2+0,, 3+4)=(-1,, 2,, 7).,

Euclidean vectors and affine vectors

In the geometrical and physical settings, sometimes it is possible to associate, in a natural way, a length to vectors as well as the notion of an angle between two vectors. When the length of vectors is defined, it is possible to also define a dot product — a scalar-valued product of two vectors — which gives a convenient algebraic characterization of both length and angle. In three-dimensions, it is further possible to define a cross product which supplies an algebraic characterization of area.


However, it is not always possible or desirable to define the length of a vector in a natural way. This more general type of spatial vector is the subject of vector spaces (for bound vectors) and affine spaces (for free vectors). In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ...


Generalizations

In more general sorts of coordinate systems, rotations of a vector (and also of tensors) can be generalized and categorized to admit an analogous characterization by their covariance and contravariance under changes of coordinates. In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... It has been suggested that this article or section be merged into Covariant transformation. ...


In mathematics, a vector is considered more than a representation of a physical quantity. In general, a vector is any element of a vector space over some field. The spatial vectors of this article are a very special case of this general definition (they are not simply any element of Rd 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. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, the domain of a function is the set of all input values to the function. ... In mathematics, the range of a function is the set of all output values produced by that function. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...


Representation of a vector

Vectors are usually denoted in boldface, as a. Other conventions include vec{a} or a, especially in handwriting. Alternately, some use a tilde (~) or a wavy underline drawn beneath the symbol, which is a convention for indicating boldface type. For the baseball player known as the Big Tilde, see Magglio Ordóñez. ...


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

Here the point 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. Image File history File links Vector_from_A_to_B.svg‎ It was done by me. ...


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

On a two-dimensional diagram, sometimes a vector perpendicular to the plane of the diagram is desired. These vectors are commonly shown as small circles. A circle with a dot at its centre indicates a vector pointing out of the front of the diagram, towards the viewer. A circle with a cross inscribed in it indicates a vector pointing into and behind the diagram. These can be thought of as viewing the tip an arrow front on and viewing the vanes of an arrow from the back. Image File history File links Notation_for_vectors_in_or_out_of_a_plane. ... Fig. ... This article is about the mathematical construct. ... An arrow is a pointed projectile that is shot with a bow. ...

A vector in the Cartesian plane, with endpoint (2,3). The vector itself is identified with its endpoint.

In order to calculate with vectors, the graphical representation may be too cumbersome. Vectors in an n-dimensional Euclidean space can be represented in a Cartesian coordinate system. The endpoint of a vector can be identified with a list of n real numbers, sometimes called a row vector or column vector. As an example in two dimensions (see image), the vector from the origin O = (0,0) to the point A = (2,3) is simply written as Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Fig. ... In linear algebra, a row vector is a 1 × n matrix, that is, a matrix consisting of a single row: The transpose of a row vector is a column vector. ... In linear algebra, a column vector is an m × 1 matrix, i. ...

overrightarrow{OA} = (2,3).

In three dimensional Euclidean space (or R3), vectors are identified with triples of numbers corresponding to the Cartesian coordinates of the endpoint (a,b,c). These numbers are often arranged into a column vector or row vector, particularly when dealing with matrices, as follows: Image File history File links Download high-resolution version (800x800, 52 KB) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Vector (spatial) ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ...

mathbf{a} = begin{bmatrix} a b c end{bmatrix}
mathbf{a} = langle a b c rangle.

Another way to express a vector in three dimensions is to introduce the three basic coordinate vectors, sometimes referred to as unit vectors:

{mathbf e}_1 = (1,0,0), {mathbf e}_2 = (0,1,0), {mathbf e}_3 = (0,0,1).

These have the intuitive interpretation as vectors of unit length pointing up the x, y, and z axis, respectively. In terms of these, any vector in R3 can be expressed in the form:

(a,b,c) = a(1,0,0) + b(0,1,0) + c(0,0,1) = a{mathbf e}_1 + b{mathbf e}_2 + c{mathbf e}_3.

Note: In introductory physics classes, these three special vectors are often instead denoted i, j, k (or boldsymbol{hat{x}}, boldsymbol{hat{y}}, boldsymbol{hat{z}} when in Cartesian coordinates), but such notation clashes with the index notation and the summation convention commonly used in higher level mathematics, physics, and engineering. This article will choose to use e1, e2, e3. Fig. ... Index notation is used in mathematics to refer to the elements of matrices or the components of a vector. ... For other topics related to Einstein see Einstein (disambig) In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate equations or formulas. ...


The use of Cartesian unit vectors boldsymbol{hat{x}}, boldsymbol{hat{y}}, boldsymbol{hat{z}} as a basis in which to represent a vector, is not mandated. Vectors can also be expressed in terms of cylindrical unit vectors boldsymbol{hat{r}}, boldsymbol{hat{theta}}, boldsymbol{hat{z}} or spherical unit vectors boldsymbol{hat{r}}, boldsymbol{hat{theta}}, boldsymbol{hat{phi}}. The latter two choices are more convenient for solving problems which possess cylindrical or spherical symmetry respectively. In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ... 2 points plotted with cylindrical coordinates The cylindrical coordinate system is a three-dimensional coordinate system which essentially extends circular polar coordinates by adding a third coordinate (usually denoted ) which measures the height of a point above the plane. ... A point plotted using the spherical coordinate system In mathematics, the spherical coordinate system is a coordinate system for representing geometric figures in three dimensions using three coordinates: the radial distance of a point from a fixed origin, the zenith angle from the positive z-axis, and the azimuth angle...


Addition and scalar multiplication

Vector equality

Two vectors are said to be equal if they have the same magnitude and direction. However if we are talking about free vectors, then two free vectors are equal if they have the same base point and end point.


For example, the vector e1 + 2e2 + 3e3 with base point (1,0,0) and the vector e1+2e2+3e3 with base point (0,1,0) are different free vectors, but the same (displacement) vector.


Vector addition and subtraction

Let a=a1e1 + a2e2 + a3e3 and b=b1e1 + b2e2 + b3e3, where e1, e2, e3 are orthogonal unit vectors (Note: they only need to be linearly independent, i.e. not parallel and not in the same plane, for these algebraic addition and subtraction rules to apply)


The sum of a and b is:

mathbf{a}+mathbf{b} =(a_1+b_1)mathbf{e_1} +(a_2+b_2)mathbf{e_2} +(a_3+b_3)mathbf{e_3}

The addition may be represented graphically by placing the start of the arrow 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:

The addition of two vectors a and b

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 free 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). Image File history File links Vector_addition. ... A parallelogram. ...


The difference of a and b is:

mathbf{a}-mathbf{b} =(a_1-b_1)mathbf{e_1} +(a_2-b_2)mathbf{e_2} +(a_3-b_3)mathbf{e_3}

Subtraction of two vectors can be geometrically defined as follows: to subtract 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 ab, as illustrated below:

The subtraction of two vectors a and b

If a and b are free 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 operation deserves the name "subtraction" because (ab) + b = a. Image File history File links Vector_subtraction. ...


Scalar multiplication

A vector may also be multiplied, or re-scaled, by a real number r. In the context of spatial vectors, these real numbers are often called scalars (from scale) to distinguish them from vectors. The operation of multiplying a vector by a scalar is called scalar multiplication. The resulting vector is: In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

rmathbf{a}=(ra_1)mathbf{e_1} +(ra_2)mathbf{e_2} +(ra_3)mathbf{e_3}
Scalar multiplication of a vector by a factor of 3 stretches the vector out.
Scalar multiplication of a vector by a factor of 3 stretches the vector out.

Intuitively, multiplying by a scalar r stretches a vector out by a factor of r. Geometrically, this can be visualized (at least in the case when r is an integer) as placing r copies of the vector in a line where the endpoint of one vector is the initial point of the next vector. Image File history File links Scalar_multiplication_by_r=3. ... Image File history File links Scalar_multiplication_by_r=3. ...


If r is negative, then the vector changes direction: it flips around by an angle of 180°. Two examples (r = -1 and r = 2) are given below:

The scalar multiplications 2a and −a of a vector a

Scalar multiplication is distributive over vector addition in the following sense: r(a + b) = ra + rb for all vectors a and b and all scalars r. One can also show that a - b = a + (-1)b. Image File history File links Scalar_multiplication_of_vectors2. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...


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. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...


In physics, scalars may also have a unit of measurement associated with them. For instance, Newton's second law is Newtons laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ...

{mathbf F} = m{mathbf a}

where F has units of force, a has units of acceleration, and the scalar m has units of mass. In one possible physical interpretation of the above diagram, the scale of acceleration is, for instance, 2 m/s² : cm, and that of force 5 N : cm. Thus a scale ratio of 2.5 kg : 1 is used for mass. Similarly, if displacement has a scale of 1:1000 and velocity of 0.2 cm : 1 m/s, or equivalently, 2 ms : 1, a scale ratio of 0.5 : s is used for time.


Length and the dot product

Length of a vector

The length or magnitude or norm of the vector a is denoted by ||a|| or, less commonly, |a|, which is not to be confused with the absolute value (a scalar "norm"). For other uses of this word, see Length (disambiguation). ... The magnitude of a mathematical object is its size: a property by which it can be larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs. ... 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. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ...


The length of the vector a = a1e1 + a2e2+ a3e3 in a three-dimensional Euclidean space, where e1, e2, e3 are orthogonal unit vectors, can be computed with the Euclidean norm Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...

left|mathbf{a}right|=sqrt{{a_1}^2+{a_2}^2+{a_3}^2}

which is a consequence of the Pythagorean theorem since the basis vectors e1 , e2 , e3 are orthogonal unit vectors. In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ...


This happens to be equal to the square root of the dot product of the vector with itself: In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ...

left|mathbf{a}right|=sqrt{mathbf{a}cdotmathbf{a}}

Vector length and units

If a vector is itself spatial, the length of the arrow depends on a dimensionless scale. If it represents e.g. a force, the "scale" is of physical dimension length/force. Thus there is typically consistency in scale among quantities of the same dimension, but otherwise scale ratios may vary; for example, if "1 newton" and "5 m" are both represented with an arrow of 2 cm, the scales are 1:250 and 1 m:50 N respectively. Equal length of vectors of different dimension has no particular significance unless there is some proportionality constant inherent in the system that the diagram represents. Also length of a unit vector (of dimension length, not length/force, etc.) has no coordinate-system-invariant significance. In dimensional analysis, a dimensionless number (or more precisely, a number with the dimensions of 1) is a pure number without any physical units. ... A scale is either a device used for measurement of weights, or a series of ratios against which different measurements can be compared. ... Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ... The word proportionality may have one of a number of meanings: In mathematics, proportionality is a mathematical relation between two quantities. ...


Unit vector

Main article: Unit vector a.k.a. Direction vector

A unit vector is any vector with a length of one; geometrically, it indicates a direction but no magnitude. If you have a vector of arbitrary length, you can divide it by its length to create a unit vector. This is known as normalizing a vector. A unit vector is often indicated with a hat as in â. In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ... In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ...

The normalization of a vector a into a unit vector â

To normalize a vector a = [a1, a2, a3], scale the vector by the reciprocal of its length ||a||. That is: Image File history File links Vector_normalization. ...

mathbf{hat{a}} = frac{mathbf{a}}{left|mathbf{a}right|} = frac{a_1}{left|mathbf{a}right|}mathbf{e_1} + frac{a_2}{left|mathbf{a}right|}mathbf{e_2} + frac{a_3}{left|mathbf{a}right|}mathbf{e_3}

Dot product

Main article: Dot product

The dot product of two vectors a and b (sometimes called the inner product, or, since its result is a scalar, the scalar product) is denoted by a ∙ b and is defined as: In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

mathbf{a}cdotmathbf{b} =left|mathbf{a}right|left|mathbf{b}right|costheta

where ||a|| and ||b|| denote the norm (or length) of a and b, and θ is the measure of the angle between 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. In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ... This article is about angles in geometry. ... In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ...


The dot product can also be defined as the sum of the products of the components of each vector:

mathbf{a} cdot mathbf{b} = langle a_1, a_2, dots, a_n rangle cdot langle b_1, b_2, dots, b_n rangle = a_1 b_1 + a_2 b_2 + dots + a_n b_n

where a and b are vectors of n dimensions; a1, a2, …, an are coordinates of a; and b1, b2, …, bn are coordinates of b.


This operation is often useful in physics; for instance, work is the dot product of force and displacement. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Lightning is the electric breakdown of air by strong electric fields, producing a plasma, which causes an energy transfer from the electric field to heat, mechanical energy (the random motion of air molecules caused by the heat), and light. ... In physics, a net force acting on a body causes that body to accelerate; that is, to change its velocity. ... In Newtonian mechanics, displacement is the vector that specifies the position of a point or a particle in reference to an origin or to a previous position. ...


Cross product

Main article: Cross product

The cross product (also called the vector product or outer product) differs from the dot product primarily in that the result of the 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 the seven dimensional cross product is similar in some respects. The cross product, denoted a × b, is a vector perpendicular to both a and b and is defined as: For the cross product in algebraic topology, see Künneth theorem. ... In mathematics, the seven dimensional cross product is a generalization of the three dimensional cross product. ...

mathbf{a}timesmathbf{b} =left|mathbf{a}right|left|mathbf{b}right|sin(theta),mathbf{n}

where θ is the measure of the angle between 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. Fig. ...

An illustration of the cross product.
An illustration of the cross product.

The vector basis e1, e2 , e3 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 the figure on the right. -from the creater, wshun 02:20, 24 Dec 2004 (UTC) File links The following pages link to this file: Vector (spatial) Cross product Categories: FAL images ... -from the creater, wshun 02:20, 24 Dec 2004 (UTC) File links The following pages link to this file: Vector (spatial) Cross product Categories: FAL images ...


The cross product a × b is defined so that a, b, and a × b also becomes a right handed system (but note that a and b are not necessarily orthogonal). This is the right-hand rule. The left-handed orientation is shown on the left, and the right-handed on the right. ...


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) is not really a new operator, but a way of applying the other two multiplication operators to three vectors. The scalar triple product is sometimes denoted by (a b c) and defined as:

(mathbf{a} mathbf{b} mathbf{c}) =mathbf{a}cdot(mathbf{b}timesmathbf{c}).

It has three primary uses. First, the absolute value of the box product is the volume of the parallelepiped which has edges that are defined by the three vectors. Second, the scalar triple product is zero if and only if the three vectors are linearly dependent, which can be easily proved by considering that in order for the three vectors to not make a volume, they must all lie in the same plane. Third, the box product is positive if and only if the three vectors a, b and c are right-handed. In geometry, a parallelepiped (now usually pronounced , traditionally[1] in accordance with its etymology in Greek παραλληλ-επίπεδον, a body having parallel planes) is a three-dimensional figure like a cube, except that its faces are not squares but parallelograms. ... In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. ...


In components ( with respect to a right-handed orthonormal basis), if the three vectors are thought of as rows (or columns, but in the same order), 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: In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ...

(mathbf{a} mathbf{b} mathbf{c}) =(mathbf{c} mathbf{a} mathbf{b})
=(mathbf{b} mathbf{c} mathbf{a})
=-(mathbf{a} mathbf{c} mathbf{b})
=-(mathbf{b} mathbf{a} mathbf{c})
=-(mathbf{c} mathbf{b} mathbf{a})

Vector components

Illustration of tangential and normal components of a vector to a surface.
Illustration of tangential and normal components of a vector to a surface.

A component of a vector is the influence of that vector in a given direction. [1] Components are themselves vectors. Image File history File links Size of this preview: 619 × 599 pixelsFull resolution (1314 × 1272 pixel, file size: 57 KB, MIME type: image/png) Made by myself with matlab. ... Image File history File links Size of this preview: 619 × 599 pixelsFull resolution (1314 × 1272 pixel, file size: 57 KB, MIME type: image/png) Made by myself with matlab. ...


A vector is often described by a fixed number of components that sum up into this vector uniquely and totally. When used in this role, the choice of their constituting directions is dependent upon the particular coordinate system being used, such as Cartesian coordinates, spherical coordinates or polar coordinates. For example, axial component of a vector is such its component whose direction is determined by one of the Cartesian coordinate axes, whereas radial and tangential components relate to the radius of rotation of an object as their direction of reference. The former is parallel to the radius and the latter is orthogonal to it. [2] Both remain orthogonal to the axis of rotation at all times. (In two dimensions this requirement becomes redundant as the axis degenerates to a point of rotation.) The choice of a coordinate system doesn't affect properties of a vector or its behaviour under transformations. Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... A coordinate axis is one of a set of vectors that defines a coordinate system. ... Illustration of tangential and normal components of a vector to a surface. ... This article is about an authentication, authorization, and accounting protocol. ... This article is about rotation as a movement of a physical body. ... Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. ... Fig. ... Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ... In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class. ...


Vectors as directional derivatives

A vector may also be defined as a directional derivative: consider a function f(xα) and a curve xα(σ). Then the directional derivative of f is a scalar defined as In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P, in the direction of V. It therefore generalizes the notion of a partial derivative, in which the... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...


frac{df}{dsigma} = frac{dx^alpha}{dsigma}frac{partial f}{partial x^alpha}.


where the index α is summed over the appropriate number of dimensions (e.g. from 1 to 3 in 3-dimensional Euclidian space, from 0 to 3 in 4-dimensional spacetime, etc.). Then consider a vector tangent to xα(σ): For other topics related to Einstein see Einstein (disambig) In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate equations or formulas. ...


t^alpha = frac{dx^alpha}{dsigma}.


We can rewrite the directional derivative in differential form (without a given function f) as


frac{d}{dsigma} = t^alphafrac{partial}{partial x^alpha}.


Therefore any directional derivative can be identified with a corresponding vector, and any vector can be identified with a corresponding directional derivative. We can therefore define a vector precisely:

mathbf{a} equiv a^alpha frac{partial}{partial x^alpha}.

References

  1. ^ Indeed in Latin the word vector means "one who carries"; Latin veho = "I carry". For historical development of the word vector, see "vector n.". Oxford English Dictionary. Oxford University Press. 2nd ed. 1989.. See also Jeff Miller. Earliest Known Uses of Some of the Words of Mathematics. Retrieved on 2007-05-25. here the vector is what would carry a point from A to B.

Mathematical treatments of spatial vectors Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 145th day of the year (146th in leap years) in the Gregorian calendar. ...

  • Apostol, T. (1967). Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra. John Wiley and Sons. ISBN 978-0471000051. 
  • Apostol, T. (1969). Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications. John Wiley and Sons. ISBN 978-0471000075. 
  • Pedoe, D. (1988). Geometry: A comprehensive course. Dover. ISBN 0-486-65812-0. .

Physical treatments Tom Mike Apostol (born 1923) is an analytic number theorist who teaches at California Institute of Technology. ... Dan Pedoe (1910 to 1998) was an English-born mathematician and geometer with a career spanning more than sixty years. ...

  • Aris, R. (1990). Vectors, Tensors and the Basic Equations of Fluid Mechanics. Dover. ISBN 978-0486661100. 
  • Feynman, Leighton, Sands (2005). "Chapter 11", The Feynman Lectures on Physics, Volume I, 2nd ed, Addison Wesley. ISBN 978-0805390469. 

Cover of the book on quantum mechanics The Feynman Lectures on Physics, by Richard Feynman, Robert Leighton, and Matthew Sands is perhaps Feynmans most accessible technical work, and is considered a classic introduction to modern physics, including lectures on mathematics, electromagnetism, Newtonian physics, quantum physics, and even the relation...

See also

In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... In relativity, a four-vector is a vector in a four-dimensional real vector space, whose components transform like the space and time coordinates (ct, x, y, z) under spatial rotations and boosts (a change by a constant velocity to another inertial reference frame). ... Two-dimensional analogy of space-time curvature described in General Relativity. ... A normal vector is a vector which is perpendicular to a surface or manifold. ... The term null vector can have two different meanings: null vector (vector space) null vector (Minkowski space) This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation (a transformation that can be expressed as an inversion followed by a proper rotation). ... In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ... Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... For information on vectors as a mathematical object see vector (spatial). ... Illustration of tangential and normal components of a vector to a surface. ...

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  Results from FactBites:
 
Vector (spatial) - Wikipedia, the free encyclopedia (2834 words)
A spatial 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 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 length or magnitude or norm of the vector a is denoted by
  More results at FactBites »

 
 

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