|
In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but in three dimensions gains an additional sign flip under an improper rotation such as a reflection. Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image. This is as opposed to a ''true'' or ''polar'' vector, which on reflection matches its mirror image. In three dimensions the pseudovector p is associated with the cross product of two polar vectors a and b:〔 〕 : The vector p calculated this way is a pseudovector. One example is the normal to an oriented plane. An oriented plane can be defined by two non-parallel vectors, a and b,〔 (RP Feynman: §52-5 Polar and axial vectors ) from Chapter 52: Symmetry and physical laws, in: Feynman Lectures in Physics, Vol. 1 〕 which can be said to span the plane. The vector is a normal to the plane (there are two normals, one on each side – the right-hand rule will determine which), and is a pseudovector. This has consequences in computer graphics where it has to be considered when transforming surface normals. A number of quantities in physics behave as pseudovectors rather than polar vectors, including magnetic field and angular velocity. In mathematics pseudovectors are equivalent to three-dimensional bivectors, from which the transformation rules of pseudovectors can be derived. More generally in ''n''-dimensional geometric algebra pseudovectors are the elements of the algebra with dimension , written Λ''n''−1R''n''. The label 'pseudo' can be further generalized to pseudoscalars and pseudotensors, both of which gain an extra sign flip under improper rotations compared to a true scalar or tensor. ==Physical examples== Physical examples of pseudovectors include magnetic field, torque, vorticity, and the angular momentum. Consider the pseudovector 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" (viewed as an ordinary vector) points to the right, but the ''actual'' angular momentum vector of the wheel (which is still turning forward in the reflection) still points to the left, corresponding to the extra minus sign in the reflection of a pseudovector. The distinction between vectors and pseudovectors becomes important in understanding the effect of symmetry on the solution to physical systems. Consider an electric current loop in the plane that inside the loop generates a magnetic field oriented in the ''z'' direction. This system is symmetric (invariant) under mirror reflections through this plane, with the magnetic field unchanged by the reflection. But reflecting the magnetic field as a vector through that plane would be expected to reverse it; this expectation is corrected by realizing that the magnetic field is a pseudovector, with the extra sign flip leaving it unchanged. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「pseudovector」の詳細全文を読む スポンサード リンク
|