Torque as a Bivector.
Torque is generally treated in introductory physics as a vector, but this supposed nicety is only an isomorphism in three dimensional space. The bivector formalism of torque is arguably more intuitive and generalizes decidedly better to dynamics of arbitrary dimension.
Torque is a generalized force (a la Lagrangian mechanics) most aptly suited for considerations of virtual displacements occuring about some well-defined axis (that is, rotational displacements). Most introductory treatments define torque as the cross products of arbitrary forces and the vector displacement from some rotational axis in three dimensions. As is well known, the cross product is only well-defined in specific sets of dimensions, three dimensions being in that set. Thus, the proper generalization of this formalism is that of the wedge product, which results in a bivector quantity for torque. This will be shown to have a more intuitive geometrical model as well as being better suited for dynamical considerations.
Torque as a Cross Product
Torque, \(\mathbf{\tau}\), is generally introduced as a three dimensional vector defined by the cross product of two other vectors: that of force, \(\mathbf{F}\); and that of the vector between the force's point of action and some arbitrary axis of rotation, \(\mathbf{r}\). This gives us the relation:
$$ \mathbf{\tau}=\mathbf{F}\times\mathbf{r} $$
Note that while the choice of rotational axis is arbitrary, the concept of torque is really most useful only for instances where \(\mathbf{r}\) and \(\mathbf{F}\) remain perpendicular, as is the case in circular motion.
Shortcomings of the Cross Product Formalism
Perhaps the most immediate response of someone introduced to torque in this way is: "It's pointing the wrong way!". And indeed, due to this misdirected nature, there are several shortomings of this formalism both conceptually and as a mathematical implementation of generalized force.
As depicted above, real instances of this conceptualization suffer from exactly this problem of being in the wrong direction. Besides, neglecting to define the potentially new wedge product may save some marginal amount of exposition in introduction, but students are still inundated with the right-hand rule, an arguably worse allocation of both teaching time and mental resources.
Torque defined in this way suffers not only conceptual faults but an explicit fault in it's implementation as a generalized force. If we look at the definition of generalized force, we see clearly it's designed to facilitate the calculation of virtual work via the dot product with relevent virtual displacements. In instances of rotational motion, those virtual displacements under consideration are those of angular displacement, happening in a certain direction in some plane of three dimensional space. However, if torque is defined to be a vector such that it's perpendicular to the plane rotation, this dot product and thus it's virtual work should always be zero! This is a spectacular failure of the cross product formalism, and also perhaps the greatest.
Torque as a Wedge Product
Now let's consider the alternative defintion of torque as a product of vectors. We simply replace the cross product with the wedge product and are left with the following relation.
$$ \mathbf{\tau}=\mathbf{F}\wedge\mathbf{r} $$
The wedge product of two perpendicular vectors is a bivector and can be geometrically intuited as an oriented area made by the parallelagram of the two vectors. The area is then the scalar quantity associated with the torque, that is, how 'much' torque is contributed to the system. While the orientation can be described either with one of two perpendicular vectors (ew), or *gasp*, unambiguously as a direction of rotation in the plane of rotation!
Not only does this formalism aid in the conceptual maladaptions of the cross product, but the dot product between the torque and angular displacements in the plane of rotation is no longer zero. Thus, this definition is the proper way to define torque in a meaningful way, as it facilitates the calculation of virtual work which is the purpose of any generalized force.
Note that other concepts in physics also benefit from a bivector-inclusive model, such as magnetic fields.
