Everything about Vector Spatial totally explained
A
spatial vector, or simply
vector, is a geometric object which 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, pseudovectors, and transformations
An alternative characterization of spatial vectors, especially in physics, describes vectors as lists of quantities which behave a certain way under a
coordinate transformation. A
vector is required to have components that "transform like the coordinates" under
coordinate rotations. In other words, if all of space were rotated, the vector would rotate in exactly the same way. Mathematically, 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 must be similarly transformed via
v′ =
Rv. This important requirement is what distinguishes a spatial vector from any other triplet of physically meaningful quantities. For example, if
v consists of the
x,
y, and
z-components of
velocity, then
v is a vector because the components of the velocity transform under coordinate changes. On the other hand, for instance, a triplet consisting of the length, width, and height of a rectangular box could be regarded as the three components of an
abstract vector, but not a
spatial vector, since rotating the box doesn't correspondingly transform these three components. Examples of vectors include
displacement,
velocity,
electric field,
momentum,
force, and
acceleration.
In the language of
differential geometry, the requirement that the components of a vector transform according to the same matrix of the coordinate transition is equivalent to defining a
vector to be a
tensor of
contravariant rank one. However, in differential geometry and other areas of
mathematics such as
representation theory, the "coordinate transitions" need not be restricted to rotations. Other notions of spatial vector correspond to different choices of
symmetry group.
As a particular case where the symmetry group is important, all of the above examples are vectors which "transform like the coordinates" under both
proper and
improper rotations. An example of an improper rotation is a
mirror reflection. That is, these 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. Vectors with this property are called
true vectors, or
polar vectors. 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. Other examples of pseudovectors include
magnetic field,
torque, or more generally any cross product of two (true) vectors.
This distinction between vectors and pseudovectors is often ignored, but it becomes important in studying
symmetry properties. See
parity (physics).
Further Information
Get more info on 'Vector Spatial'.
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