In physics and engineering, a vector is a physical entity which has a magnitude which is a scalar (a physical quantity expressed as the product of a numerical value and a physical unit, not just a number). In contrast with a scalar, a vector has also a direction. Although it is often described by a number of "components", each of which is dependent upon the particular coordinate system being used, in physics a vector is, like a scalar, a "real" object with properties which are independent of the coordinate system used to describe it. The willingness to question previously held truths and search for new answers resulted in a period of major scientific advancements, now known as the Scientific Revolution. ... Licensure and Qualifications for the Practice of Engineering The Engineers Ring The Ritual of the Calling of an Engineer Engineering Disasters and Learning from Failure American Society of Engineering Education (ASEE) ASEE engineering profile (2003) PDF Categories: Architecture and engineering occupations | Engineering ... In science, magnitude refers to the numerical size of something: see orders of magnitude. ... Scalar is a concept that has meaning in mathematics, physics, and computing. ... A number is an abstract entity used originally to describe quantity. ... In physics and metrology, units are standards for measurement of physical quantities that need clear definitions to be useful. ...

The word vector is also now used for more general concepts (see also vector and generalizations below), but this article describes the original spatial meaning except where otherwise noted. The word vector means carrier in Latin; it is derived from the Latin verb vehere, which means to carry. ...

A common example of a vector is force — it has a magnitude and an orientation in three dimensions (or however many spatial dimensions one has), and multiple forces sum according to the parallelogram law. In physics, force is defined as the time derivative of momentum: F = dp/dt = d(m·v)/dt where F is the force (a vector quantity), p is the momentum, v is the velocity, and m is the mass. ... The parallelogram law in elementary geometry In elementary geometry, the parallelogram law states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. ...

A vector can be formally defined by its relationship to the spatial coordinate system under rotations. Alternatively, it can be defined in a coordinate-free fashion via a tangent space of a three-dimensional manifold in the language of differential geometry. These definitions are discussed in more detail below. See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... This is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...

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 are the building blocks of vector fields and vector calculus. In mathematics, a tensor is a certain kind of geometrical entity, or alternatively generalized quantity. The tensor concept includes the ideas of scalar, vector and linear operator. ... 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). ... Albert Einsteins theory of relativity is a set of two scientific theories in physics: special relativity and general relativity. ... A p-vector is the tensor obtained by taking linear combinations of the wedge product of p tangent vectors. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in 2 or more dimensions. ...

Definitions

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. Examples are "moving north at 90 km/h" or "pulling towards the center of Earth with a force of 70 newtons". In science, magnitude refers to the numerical size of something: see orders of magnitude. ... In physics, the newton (symbol: N) is the SI unit of force, named after Sir Isaac Newton in recognition of his work on classical mechanics. ...

The notion of having a "magnitude" and "direction" is formalized by saying that the vector has components that transform like the coordinates under rotations. That is, if the coordinate system undergoes a rotation described by a rotation matrix R, so that a coordinate vector x is transformed to x' = Rx, then any other vector v is similarly transformed via v' = Rv.

More generally, a vector is a tensor of contravariant rank one. In differential geometry, the term vector usually refers to quantities that are closely related to tangent spaces of a differentiable manifold (assumed to be three-dimensional and equipped with a positive definite Riemannian metric). (A four-vector is a related concept when dealing with a 4 dimensional spacetime manifold in relativity.) In mathematics, a tensor is a certain kind of geometrical entity, or alternatively generalized quantity. The tensor concept includes the ideas of scalar, vector and linear operator. ... This is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ... In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... 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). ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ... This is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ... Albert Einsteins theory of relativity is a set of two scientific theories in physics: special relativity and general relativity. ...

Examples of vectors include displacement, velocity, electric field, momentum, force, and acceleration. The term displacement can have one of several meanings, depending on context: Displacement (distance), a physical quantity in kinematics Particle displacement, acoustics of sound in air Displacement (fluid), a different physical quantity, used in fluid mechanics and navigation; used as a measure of a ships size Engine displacement, a... Velocity (symbol: v) is a vector measurement of the rate and direction of motion. ... In physics, an electric field or E-field is an effect produced by an electric charge that exerts a force on charged objects in its vicinity. ... In physics, momentum is a physical quantity related to the velocity and mass of an object. ... In physics, a net force acting on a body causes that body to accelerate; that is, to change its velocity. ... Acceleration is the time rate of change of velocity, and at any point on a v-t graph, it is given by the gradient of the tangent to that point In physics, acceleration (symbol: a) is defined as the rate of change (or time derivative) of velocity. ...

Vectors can be contrasted with scalar quantities such as distance, speed, energy, time, temperature, charge, power, work, and mass, which have magnitude, but no direction (they are invariant under coordinate rotations). The magnitude of any vector is a scalar. Scalar is a concept that has meaning in mathematics, physics, and computing. ... personal space, proxemics. ... Speed (symbol: v) is the rate of motion, or equivalently the rate of change of position, expressed as distance d moved per unit of time t. ... 8:17 am, August 6, 1945, Japanese time. ... Temperature is the physical property of a system which underlies the common notions of hot and cold; the material with the higher temperature is said to be hotter. ... Charge is a word with many different meanings. ... // Mechanical power In physics, power (symbol: P) is the amount of work W done per unit of time t. ... Energy is a fundamental quantity that every physical system possesses. ... Mass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. ... Real numbers The magnitude of a real number is usually called the absolute value or modulus. ...

A related concept is that of a pseudovector (or axial vector). This is a quantity that transforms like a vector under proper rotations, but gains an additional sign flip under improper rotations. Examples of pseudovectors include magnetic field, torque, and angular momentum. (This distinction between vectors and pseudovectors is often ignored, but it becomes important in studying symmetry properties.) To distinguish from pseudo/axial vectors, an ordinary vector is sometimes called a polar vector. 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 geometry, an improper rotation is the combination of an ordinary rotation of three-dimensional Euclidean space, that keeps the origin fixed, with a coordinate inversion (a vector x goes to −x). ... Current flowing through a wire produces a magnetic field (M) around the wire. ... The concept of torque in physics, also called moment or couple, originated with the work of Archimedes on levers. ... In physics, angular momentum intuitively measures how much the linear momentum is directed around a certain point called the origin; the moment of momentum. ...

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

Generalizations

In mathematics, a vector 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 R^d in d dimensions), which includes a variety of mathematical objects (algebras, the set of all functions from a given domain to a given linear range, and linear transformations). Note that under this definition, a tensor is a special vector! Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... 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 are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... 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 well-defined collection of things considered as a whole. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, the domain of a function is the set of all input values to the function. ... The range of a vehicle is the maximum distance it can cover without needing to be refueled or recharged. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In mathematics, a tensor is a certain kind of geometrical entity, or alternatively generalized quantity. The tensor concept includes the ideas of scalar, vector and linear operator. ...

Representation of a vector

Symbols standing for vectors are usually printed in boldface as a; this is also the convention adopted in this encyclopedia. Other conventions include or a, especially in handwriting. Alternately, some use a tilde (~) placed under the vector. The length or magnitude or norm of the vector a is denoted by |a|. The tilde is a grapheme which has several uses, described below. ... In general English usage, length (symbols: l, L) is but one particular instance of distance – an objects length is how long the object is – but in the physical sciences and engineering, the word length is in some contexts used synonymously with distance. Height is vertical distance; width (or breadth... Real numbers The magnitude of a real number is usually called the absolute value or modulus. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ...

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. File links The following pages link to this file: Vector (spatial) Categories: GFDL images ...

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; e.g "1 newton" and "5 m" are both represented with an arrow of 2cm; the scales are 1:250 and 1m:50N respectively. Equal length of vectors of different dimension has no particular significance. 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; it does not change if one alters ones system of units of measurement, for example from English units to metric 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. ...

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

In order to calculate with vectors, the graphical representation is too cumbersome. Vectors in a n-dimensional Euclidean space can be represented as a linear combination of n mutually perpendicular unit vectors. In this article, we will consider R³ as an example. In R³, we usually denote the unit vectors parallel to the x-, y- and z-axes by i, j and k respectively. Any vector a in R³ can be written as a = a₁i + a₂j + a₃k with real numbers a₁, a₂ and a₃ which are uniquely determined by a. Sometimes a is then also written as a 3-by-1 or 1-by-3 matrix: For the square matrix section, see square matrix. ...

even though this notation suppresses the dependence of the coordinates a₁, a₂ and a₃ on the specific choice of coordinate system i, j and k.

Length of a vector

The length of the vector a = a₁i + a₂j + a₃k can be computed with the Euclidian norm 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. ...

which is a consequence of the Pythagorean theorem. The Pythagorean theorem : a2 + b2 = c2 In mathematics, the Pythagorean theorem or Pythagoras theorem is a relation in Euclidean geometry between the three sides of a right triangle. ...

Vector equality

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

For example, the vector i + 2j + 3k with base point (1,0,0) and the vector i+2j+3k with base point (0,1,0) are different bound vectors, but the same (unbounded) vector.

Vector addition and subtraction

Let a=a₁i + a₂j + a₃k and b=b₁i + b₂j + b₃k.

The sum of a and b is:

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:

This addition method is sometimes called the parallelogram rule because a and b form the sides of a parallelogram and a + b is one of the diagonals. If a and b are bound vectors, then the addition is only defined if a and b have the same base point, which will then also be the base point of a + b. One can check geometrically that a + b = b + a and (a + b) + c = a + (b + c). Example of vector addition made in Appleworks and converted to . ... A parallelogram is a four-sided plane figure that has two sets of opposite parallel sides. ...

The difference of a and b is:

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 a - b, as illustrated below:

If a and b are bound vectors, then the subtraction is only defined if they share the same base point which will then also become the base point of their difference. This operation deserves the name "subtraction" because (a - b) + b = a. An example of vector subtraction made by BlackGriffen File links The following pages link to this file: Vector (spatial) Categories: GFDL images ...

In physics, vectors of different physical dimension may occur in the same diagram. However, adding or subtracting them (graphically or otherwise) is meaningless.

Scalar multiplication

A vector may also be multiplied by a real number r. In mathematics numbers are often called scalars to distinguish them from vectors, and this operation is therefore called scalar multiplication. The resulting vector is: In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

The length of ra is |r||a|. If the scalar is negative, it also changes the direction of the vector by 180^o. Two examples (r = -1 and r = 2) are given below:

Here it is important to check that the scalar multiplication is compatible with vector addition in the following sense: r(a + b) = ra + rb for all vectors a and b and all scalars r. One can also show that a - b = a + (-1)b. An example of scalar multiplication of vectors, made by BlackGriffen. ...

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. A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...

In physics, scalars also have a unit. The scale of acceleration in the diagram is e.g. 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.

Unit vector

Main article: Unit vector In mathematics, a unit vector in a normed vector space is a vector (most commonly a spatial vector) whose length is 1. ...

A unit vector is any vector with a length of one. If you have a vector of arbitrary length, you can use it to create a unit vector. This is known as normalizing a vector.

To normalize a vector a = [a₁, a₂, a₂], scale the vector by the inverse of its length ||a||. That is: Unit Vector File links The following pages link to this file: Vector (spatial) ...

Dot product

Main article: Dot product In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ...

The dot product of two vectors a and b (sometimes called inner product, or, since its result is a scalar, the scalar product) is denoted by a·b and is defined as: In mathematics, an inner product space is a vector space with additional structure, an inner product, scalar product or dot product, which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

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. This operation is often useful in physics; for instance, work is the dot product of force and displacement. 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 are functions of an angle, important when studying triangles and modeling periodic phenomena. ... The willingness to question previously held truths and search for new answers resulted in a period of major scientific advancements, now known as the Scientific Revolution. ... Energy is a fundamental quantity that every physical system possesses. ... 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 one of two subtly different quantities measuring distance and direction. ...

Cross product

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

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. Which vector is the correct one depends upon the orientation of the vector space, i.e. on the handedness of the coordinate system. The coordinate system i, j, k is called right handed, if the three vectors are situated like the thumb, index finger and middle finger (pointing straight up from your palm) of your right hand. Graphically the cross product can be represented by this figure A human hand typically has four fingers and a thumb The hand (med. ...

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

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

Scalar triple product

The scalar triple product (also called the box product or mixed triple product) isn't really a new operator, but a way of applying the other two multiplication operators to three vectors. The scalar triple product is denoted by (a b c) and defined as:

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 oriented like the coordinate system i, j and k. In geometry, a parallelepiped or parallelopipedon is a 3-dimensional figure that can be thought of as an (impractical) box that, while not necessarily having right angles, still has its upper surface level whenever it rests its lower surface on something level. ... In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ...

In coordinates, if the three vectors are thought of as rows, the scalar triple product is simply the determinant of the 3-by-3 matrix having the three vectors as rows. The scalar triple product is linear in all three entries and anti-symmetric in the following sense: In linear algebra, the determinant is a function that associates a scalar det(A) to every square matrix A. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

Technically, the scalar triple product isn't a scalar, it is a pseudoscalar: under a coordinate inversion (x goes to −x), it flips sign. Scalar is a concept that has meaning in mathematics, physics, and computing. ...

External links

• Online vector identities (pdf)
• Vectors at Wikibooks