## Vector (spatial)

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Message 1 of 18 , Dec 1, 2007
Vector (spatial)
(Redirected from Vector sum)
spatial coordinates. For a generalization, see vector space. For
other uses, see vector.

A vector going from A to B.A spatial vector, or simply vector, is a
concept characterized by a magnitude and a direction, and which sums
with other vectors according to the Parallelogram law. 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

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.

Contents [hide]
1 Overview
1.1 Use in physics and engineering
1.2 Vectors in Cartesian space
1.3 Euclidean vectors and affine vectors
1.4 Generalizations
2 Representation of a vector
3.1 Vector equality
3.3 Scalar multiplication
4 Length and the dot product
4.1 Length of a vector
4.1.1 Vector length and units
4.2 Unit vector
4.3 Dot product
5 Cross product
5.1 Scalar triple product
6 Vector components
7 Vectors as directional derivatives
8 Pseudovectors
9 References

 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.

 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 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.

 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.

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

 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

 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 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.

 Representation of a vector
Vectors are usually denoted in boldface, as a. Other conventions
include 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.

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.

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.

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

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:

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

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:

Note: In introductory physics classes, these three special vectors
are often instead denoted i, j, k (or 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.

The use of Cartesian unit vectors as a basis in which to represent
a vector, is not mandated. Vectors can also be expressed in terms of
cylindrical unit vectors or spherical unit vectors . The latter two
choices are more convenient for solving problems which possess
cylindrical or spherical symmetry respectively.

 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.

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:

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 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).

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 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 (a − b) + b = a.

 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:

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.

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:

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.

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 physics, scalars may also have a unit of measurement associated
with them. For instance, Newton's second law is

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").

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

which is a consequence of the Pythagorean theorem since the basis
vectors e1 , e2 , e3 are orthogonal unit vectors.

This happens to be equal to the square root of the dot product of
the vector with itself:

 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.

 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.

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

 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:

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.

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

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.

 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:

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.

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.

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

The cross product of two vectors is a pseudovector (see below).

 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:

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 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:

 Vector components

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.

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,
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.

 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

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):

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

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:

 Pseudovectors
Main article: pseudovector
Many vectors are defined in such a way that, if all of space were
flipped around through a mirror (or otherwise subjected to an
improper rotation), that vector would flip around in exactly the
same way. These are called true vectors, or polar vectors, and
include such things as a particle's position and velocity (in
physics), or a vector defined as a directional derivative (see
above). However, other vectors are defined in such a way that, upon
flipping through a mirror, the vector flips in the same way, but
also acquires a negative sign. These are called pseudovectors (or
axial vectors), and most commonly occur as cross products of true
vectors.

One example of an axial vector is angular momentum. Driving in a
car, and looking forward, each of the wheels has an angular momentum
vector pointing to the left. If the world is reflected in a mirror
which switches the left and right side of the car, the reflection of
this angular momentum vector points to the right, but the actual
angular momentum vector of the wheel still points to the left,
corresponding to the minus sign.
• A spatial vector, or simply vector, is a geometric object that has both a magnitude and a direction. A vector is frequently represented by a line segment
Message 2 of 18 , Jul 3, 2008
A spatial vector, or simply vector, is a geometric object that has
both a magnitude and a direction. A vector is frequently represented
by a line segment connecting the initial point A with the terminal
point B and denoted

The magnitude is the length of the segment and the direction
characterizes the displacement of B relative to A: how much one
should move the point A to "carry" it to the point B.[1]

Many algebraic operations on real numbers have close analogues for
vectors. Vectors can be added, subtracted, multiplied by a number,
and flipped around so that the direction is reversed. These
operations obey the familiar algebraic laws: commutativity,
associativity, distributivity. The sum of two vectors with the same
initial point can be found geometrically using the parallelogram
law. Multiplication by a positive number, commonly called a scalar
in this context, amounts to changing the magnitude of vector, that
is, stretching or compressing it while keeping its direction;
multiplication by -1 preserves the magnitude of the vector but
reverses its direction.

Cartesian coordinates provide a systematic way of describing vectors
and operations on them. A vector becomes a tuple of real numbers,
its scalar components. Addition of vectors and multiplication of a
vector by a scalar are simply done component by component, see
coordinate vector.

Vectors play an important role in physics: velocity and acceleration
of a moving object and forces acting on a body are all described by
vectors. Many other physical quantities can be usefully thought of
as vectors. One has to keep in mind, however, that the components of
a physical vector depend on the coordinate system used to describe
it. Other vector-like objects that describe physical quantities and
transform in a similar way under changes of the coordinate system
include pseudovectors and tensors.
• A spatial vector, or simply vector, is a geometric object that has both a magnitude and a direction. A vector is frequently represented by a line segment
Message 3 of 18 , Jul 6, 2008
A spatial vector, or simply vector, is a geometric object that has
both a magnitude and a direction. A vector is frequently represented
by a line segment connecting the initial point A with the terminal
point B and denoted

The magnitude is the length of the segment and the direction
characterizes the displacement of B relative to A: how much one
should move the point A to "carry" it to the point B.[1]

Many algebraic operations on real numbers have close analogues for
vectors. Vectors can be added, subtracted, multiplied by a number,
and flipped around so that the direction is reversed. These
operations obey the familiar algebraic laws: commutativity,
associativity, distributivity. The sum of two vectors with the same
initial point can be found geometrically using the parallelogram law.
Multiplication by a positive number, commonly called a scalar in this
context, amounts to changing the magnitude of vector, that is,
stretching or compressing it while keeping its direction;
multiplication by -1 preserves the magnitude of the vector but
reverses its direction.

Cartesian coordinates provide a systematic way of describing vectors
and operations on them. A vector becomes a tuple of real numbers, its
scalar components. Addition of vectors and multiplication of a vector
by a scalar are simply done component by component, see coordinate
vector.

Vectors play an important role in physics: velocity and acceleration
of a moving object and forces acting on a body are all described by
vectors. Many other physical quantities can be usefully thought of as
vectors. One has to keep in mind, however, that the components of a
physical vector depend on the coordinate system used to describe it.
Other vector-like objects that describe physical quantities and
transform in a similar way under changes of the coordinate system
include pseudovectors and tensors.
• A spatial vector, or simply vector, is a geometric object that has both a magnitude and a direction. A vector is frequently represented by a line segment
Message 4 of 18 , Jul 19, 2008
A spatial vector, or simply vector, is a geometric object that has
both a magnitude and a direction. A vector is frequently represented
by a line segment connecting the initial point A with the terminal
point B and denoted
• Vectors play an important role in physics: velocity and acceleration of a moving object and forces acting on a body are all described by vectors. Many other
Message 5 of 18 , Aug 8, 2008
Vectors play an important role in physics: velocity and acceleration of
a moving object and forces acting on a body are all described by
vectors. Many other physical quantities can be usefully thought of as
vectors. One has to keep in mind, however, that the components of a
physical vector depend on the coordinate system used to describe it.
Other vector-like objects that describe physical quantities and
transform in a similar way under changes of the coordinate system
include pseudovectors and tensors.
• A spatial vector, or simply vector, is a geometric object that has both a magnitude and a direction. A vector is frequently represented by a line segment
Message 6 of 18 , Aug 8, 2008
A spatial vector, or simply vector, is a geometric object that has
both a magnitude and a direction. A vector is frequently represented
by a line segment connecting the initial point A with the terminal
point B and denoted

The magnitude is the length of the segment and the direction
characterizes the displacement of B relative to A: how much one
should move the point A to "carry" it to the point B.[1]

Many algebraic operations on real numbers have close analogues for
vectors. Vectors can be added, subtracted, multiplied by a number,
and flipped around so that the direction is reversed. These
operations obey the familiar algebraic laws: commutativity,
associativity, distributivity. The sum of two vectors with the same
initial point can be found geometrically using the parallelogram law.
Multiplication by a positive number, commonly called a scalar in this
context, amounts to changing the magnitude of vector, that is,
stretching or compressing it while keeping its direction;
multiplication by -1 preserves the magnitude of the vector but
reverses its direction.

Cartesian coordinates provide a systematic way of describing vectors
and operations on them. A vector becomes a tuple of real numbers, its
scalar components. Addition of vectors and multiplication of a vector
by a scalar are simply done component by component, see coordinate
vector.
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