2017 3rd International Conference on Management Science and Innovative Education (MSIE 2017) ISBN: 978-1-60595-488-2
Physical Origin of Pseudovector Exemplified by Torque
XIAOPING QIN, PENG LI, HONGXIA WANG and ZHAOCUN ZONG
ABSTRACT
In college physics, cross product and pseudovector are always very confusing for most students. Here we explain from the bottom why cross product is defined as the mysterious determinant appeared in every textbook. Only middle-school knowledge of lever principle is required in the derivation. Furthermore, we introduce a strict definition of physical vector according to the transformation property under space rotation, and show explicitly that cross product of two real vectors is still a vector. Such vector is a pseudovector due to its special transformation property under space inversion.
KEYWORDS Cross Product, Pseudovector, Torque.
INTRODUCTION
College students often do not well understand why torque is a vector. The deepest explanation roots in the property of infinitesimal rotation [1], and involves an indispensable detour about the incommutability of successive space rotations, which is rather a lengthy story to tell and usually causes more confusion. Meanwhile, torque is formally defined by a determinant as shown in many textbooks [2]. Students were told that the cross product of two vectors is still a vector, so that torque must be a vector. But this formalism is still too abstract. Most students only accept the common sense that torque is the multiplication of the force and the arm length. Obviously, there is a mental gap between the abstract definition of torque and the plain understanding of simple lever mechanics. In this paper we explain from the bottom how the lever principle leads to the modern definition of torque both geometrically and algebraically. This connection is valid for all physical quantities defined via a cross product. Furthermore, a constraint on physical vectors according to their transformation property under space rotation is introduced and is verified for the cross product of vectors. In this approach the concept of pseudovector is well understood.
______
Xiaoping Qin, Peng Li, Hongxia Wang, Zhaocun Zong. School of Mathematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan 316022, China; Key Laboratory of Oceanographic Big Data Mining & Application of Zhejiang Province, Zhoushan 316022, China Corresponding author: Peng Li, [email protected]
326 GEOMETRIC INTERPRETATION OF CROSS PRODUCT
Torque, also quoted as the moment of force, reflects the turning effect of the force. It describes the tendency of the lever to rotate about a point or axis under the force. As shown in Fig.1, one end of a lever is hinged to point O, and a force F is applied to the other end, so that the lever is about to turn in the x-y plane. As learned in middle school, the torque of F depends on the multiplication of two factors: the force magnitude F and the arm length d. Therefore, the torque is denoted by
M Fd Fr sin , (1)
where r is the position vector from O to the point of action, and α is the angle between r and F. Since the torque tends to rotate the lever along the z direction, we can artificially assign z axis as the direction of the torque. In this spirit the torque can be tentatively described by a vector as
M r F . (2)
Here we have adopted the geometric definition of cross product, i.e., its direction is defined as right-handedly perpendicular to the multiplier vectors, and its magnitude is defined by equation (1). Moreover, the above argument still holds for r and F that do not lie in the x-y plane.
Figure 1. The lever principle.
ALGEBRAIC INTERPRETATION OF CROSS PRODUCT
The above definition of torque can be explained in an equivalent yet different approach. We still use Figure (1) as an illustration. Since F can be decomposed as Fx and Fy, we can first calculate the torque of these components separately then add them up. The magnitude of the vertical force is Fy and its arm length is x, so the torque of Fy is simply xFy. It tends to cause a rotation about the z direction. On the other hand, the horizontal force Fx has an arm length y, however, since it tends to cause a
327 rotation about the opposite direction of z axis, the corresponding torque is -yFx. Now we add up the total torque along the z direction as
Mz xF y yF x . (3)
This analysis can be generalized to other directions by simply applying the following cyclic rule
xy , yz , zx . (4)
Therefore, we find
Mx yF z zF y , and My zF x xF z . (5)
Equations (3) and (5) can be packaged into a concise form as the determinant below
ex e y e z M r F x y z , (6)
FFFx y z
which is exactly the definition of torque in standard textbooks.
VECTOR AND SPACE ROTATION
Now we look at another key question: why equation (6) defines a vector? The question is pertinent because a vector is not simply three numbers stacked together as (,,)abc . A physically legitimate vector must transform properly under space rotation, just as the position vector r= (x, y, z) does. Given a counter example, assume a, b, and c denotes the weight of three different particles, respectively. Then the quantity S= (a, b, c) does not vary under any space rotation at all! Certainly, this is not a vector, but a scalar with three independent components. The properly rotation of a vector A about a given axis can be described by a matrix operator Rn,θ, where the axis is denoted by a unit vector n=(nx, ny, nz) and the rotation angle is θ. The operator transforms the (column) vector to A' as [3-4]
0 nnzy 2 N nn0 A' = Rn, A, where RNNn,3 I sin (1 cos ) zx. (7) nnyx0
In this formalism, if the torque M is really a vector it must abide
M' r' F' Rn, M , (8)
In other words, to prove M to be a vector it suffices to prove
Rn,,, r R n F R n r F . (9)
328 Although equation (9) can be verified straightforwardly, the procedure is rather lengthy and tedious. Here we demonstrate a special rotation of angle θ about the z axis, which explains the essence of the proof. In this case
cos sin 0 R sin cos 0 z, (10) 0 0 1
such that
x x' x cos y sin FFFFx x' x cos y sin ry r' y' x sin y cos FFFFF F' ' sin cos y y x y (11) z z' z FFFz z' z
Clearly, we have
yFzFx'z ' ' y ' ( sin y cos ) FzF z ( x sin F y cos ) M x cos M y sin ,
zFxFzF'x ' ' z ' ( x cos F y sin ) ( x cos y sin ) FM z x sin M y cos ,
xFyFx'y ' ' x ' ( cos yF sin )( x sin F y cos ) ( xyF sin cos )( x cos F y sin ) M z .
Above equations can be summarized in matrix form as
MMMx x' x MMMM M' ' R y y z, y . (12) MMMz z' z
It is easy to verify that the above derivation is also valid for rotations about the x or y axes. Because any rotation can be decomposed by successive rotations about the x, y, and z axes, we have indeed proved that torque M truly abides the rotation transformation for a vector. As a generalization, the cross product of any two vectors is still a vector.
PSEUDOVECTOR AND SPACE INVERSION
Although we have shown that the cross product does define a vector that abides the proper rotational transformation, there is still a special character to address. Vectors can be divided into two categories: polar and axial, according to their transform property under space inversion. In many places, polar vectors are called real vectors, and axial vectors are called pseudovectors. Under space inversion, the bases are inversed as
eexx' , eeyy' , eezz' . (13)
329 A polar vector
AAAA e e e xx y y z z under the new bases will be denoted by
A'AAA'''''' e e e xx y y z z Since A and A' denote the same vector, we have
AAxx', AAyy', AAzz'. (14)
Therefore, loosely speaking, polar vectors change sign under space inversion. The position vector r, velocity vector v, and force F, electric field E, et al., are all polar vectors. However, for a torque defined by M = r F , the situation is different. Since both r and F change sign under inversion, we have
MMxx', MMyy', MMzz'. (15)
That is to say, the torque does not change sign under space inversion. Such vectors are called axial or pseudovectors. Besides, the angular momentum L, the magnetic dipole moment μ, the angular velocity , and the magnetic field B, are all pseudovectors. Furthermore, it is easily seen that the cross product of a real vector and a pseudovector is, however, a real vector. A well-known example is given by the Lorenz force FLorenz =q v B .
ACKNOWLEDGEMENT
The corresponding author Peng Li is supported by the Teaching Enhancement P rogram of Zhejiang Ocean University. The coauthors are supported by NSFC Grant No. 11505154, 11305141, 11104248, and 11204271. Corresponding author e-mail: Peng Li, [email protected], School of Math ematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan 31 6022, China; Key Laboratory of Oceanographic Big Data Mining & Application of Zhejiang Province, Zhoushan 316022, China
REFERENCES
1. H. Jeffreys, B. S. Jeffreys, Methods of mathematical physics. Cambridge University Press, 1999. 2. Information on: https://en.wikipedia.org/wiki/Cross_product 3. Camillo J. Taylor, David J. Kriegman, "Minimization on the Lie Group SO (3) and Related Manifolds" (PDF). Technical Report No. 9405. Yale University, 1994. 4. Information on: https://en.wikipedia.org/wiki/Rotation_matrix
330