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therefore chooses an orientation of every basis of every zero-dimensional vector space. If all zero-dimensional vector spaces are assigned this orientation, then, because all isomorphisms among zero-dimensional vector spaces preserve the ordered basis, they also preserve the orientation. This is
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110:. Thus, in three dimensions, it is impossible to make the left hand of a human figure into the right hand of the figure by applying a displacement alone, but it is possible to do so by reflecting the figure in a mirror. As a result, in the three-dimensional
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The concept of orientation degenerates in the zero-dimensional case. A zero-dimensional vector space has only a single point, the zero vector. Consequently, the only basis of a zero-dimensional vector space is the empty set
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has an attitude given by a straight line parallel to it, an orientation given by its sense (often indicated by an arrowhead) and a magnitude given by its length. Similarly, a
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1138:, which is the trivial group and thus has a single connected component; this corresponds to the canonical orientation on a zero-dimensional vector space). The
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common to these planes ), an orientation (sometimes denoted by a curved arrow in the plane) indicating a choice of sense of traversal of its boundary (its
61:, right-handed bases are typically declared to be positively oriented, but the choice is arbitrary, as they may also be assigned a negative orientation. A
446:{\displaystyle \mathbf {A} _{1}={\begin{pmatrix}\cos \alpha &-\sin \alpha &0\\\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}}
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unlike the case of higher-dimensional vector spaces where there is no way to choose an orientation so that it is preserved under all isomorphisms.
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does not have a distinguished element (i.e. a privileged basis) there is no natural choice of which component is positive. Contrast this with GL(
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restrictions, this is not always possible. A manifold that admits a smooth choice of orientations for its tangent spaces is said to be
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However, there are situations where it is desirable to give different orientations to different points. For example, consider the
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The algebraic bivector is not specific on shape; geometrically it is an amount of oriented area in a specific plane, that's all.
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The one-dimensional case deals with a line which may be traversed in one of two directions. There are two orientations to a
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The orientation of a volume may be determined by the orientation on its boundary, indicated by the circulating arrows.
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goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In
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Parallel plane segments with the same attitude, magnitude and orientation, all corresponding to the same bivector
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Every ordered basis lives in one equivalence class or another. Thus any choice of a privileged ordered basis for
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102:, the notion of orientation makes sense in arbitrary finite dimension, and is a kind of asymmetry that makes a
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682:, a zero-dimensional vector space is the same as a zero-dimensional vector space with ordered basis. Choosing
1289:{\displaystyle \pi _{0}(\operatorname {GL} (V))=(\operatorname {GL} (V)/\operatorname {GL} ^{+}(V)=\{\pm 1\}}
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whose sole member is the empty set. This means that an orientation of a zero-dimensional space is a function
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548:{\displaystyle \mathbf {A} _{2}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&-1\end{pmatrix}}}
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The ordering of elements in a basis is crucial. Two bases with a different ordering will differ by some
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choice of which direction on this line is positive. An orientation is just such a choice. Any nonzero
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determines an orientation: the orientation class of the privileged basis is declared to be positive.
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according to whether the determinant of the transformation is positive or negative (except for GL
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It is therefore possible to orient a point in two different ways, positive and negative.
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Clifford (geometric) algebras with applications to physics, mathematics, and engineering
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are "positively" oriented and which are "negatively" oriented. In the three-dimensional
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The connection of this with the determinant point of view is: the determinant of an
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The left-handed orientation is shown on the left, and the right-handed on the right.
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Geometric
Algebra for Computer Science: An Object-Oriented Approach to Geometry
1166:. These orbits are precisely the equivalence classes referred to above. Since
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is an assignment of +1 to one equivalence class and −1 to the other.
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sometimes has the selected orientation indicated by the orientation of a
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New foundations for classical mechanics: Fundamental
Theories of Physics
808:(a connected subset of a line), the two possible orientations result in
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is a broader notion that, in two dimensions, allows one to say when a
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Choice of reference for distinguishing an object and its mirror image
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can be interpreted as the induced action on the top exterior power.
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William
Anthony Granville (1904). "ยง178 Normal line to a surface".
1612: โ Physical quantity that changes sign with improper rotation
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just as there are two orientations to a circle. In the case of a
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of this permutation is ยฑ1. This is because the determinant of a
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in three dimensions has an attitude given by the family of
1606: โ Generalization of an orientation of a vector space
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is equal to the signature of the associated permutation.
215:. The property of having the same orientation defines an
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Pages displaying short descriptions of redirect targets
1490:: attitude, orientation, and magnitude. For example, a
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1416:{\displaystyle V_{n}(V)/\operatorname {GL} ^{+}(V)}
118:and left-handed (or right-chiral and left-chiral).
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319:if its determinant is positive. For instance, in
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652:{\displaystyle \{\{\emptyset \}\}\to \{\pm 1\}.}
307:be a nonsingular linear mapping of vector space
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457:Cartesian plane is not orientation-preserving:
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957:-form we can evaluate it on an ordered set of
269:on which it is built). Any choice of a linear
1732:(2nd ed.). Morgan Kaufmann. p. 32.
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662:Because there is only a single ordered basis
106:impossible to replicate by means of a simple
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1819:(2nd ed.). Springer. p. 21.
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1618: โ Ways to represent 3D rotations
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1597: โ Property in group theory
894:{\displaystyle {\tbinom {n}{k}}}
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231:determined by this relation. An
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758:fundamental theorem of calculus
1795:. Ginn & Company. p.
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152:. It is a standard result in
51:orientation of a vector space
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982:} is a privileged basis for
1850:Encyclopedia of Mathematics
1583:Cartesian coordinate system
1090:freely and transitively on
331:is orientation-preserving:
327:Cartesian axis by an angle
267:Cartesian coordinate system
156:that there exists a unique
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1650:"Vector Space Orientation"
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1122:). Note that the group GL(
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576:{\displaystyle \emptyset }
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134:real vector space and let
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1768:. Springer. p. 397.
1595:Even and odd permutations
1449:{\displaystyle \{\pm 1\}}
148:be two ordered bases for
1689:"Orientation-Preserving"
1514:Orientation on manifolds
1106:). This means that as a
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30:Not to be confused with
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323:a rotation around the
317:orientation-preserving
211:; otherwise they have
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32:Orientation (geometry)
1760:Forms and pseudoforms
1693:mathworld.wolfram.com
1687:W., Weisstein, Eric.
1654:mathworld.wolfram.com
1648:W., Weisstein, Eric.
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902:. The vector space ฮ
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1877:Analytic geometry
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1460:Geometric algebra
1317:{\displaystyle V}
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1635:References
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1296:, and the
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122:Definition
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