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I was looking at a very simplistic case of rotational dynamics yesterday, which described the in-plane rotation of a rod of length $L$ about its center of mass. The position vector of a point at the edge of the rod was defined as $$\vec{P}=\left(\begin{matrix}\frac{L}{2}\cos\theta\\ \frac{L}{2}\sin\theta\\ 0\end{matrix}\right).$$ The angular velocity was defined as $$\vec{\omega}=\left(\begin{matrix}0\\ 0\\ {\theta}'\end{matrix}\right).$$ Obviously the velocity was defined as $\vec{v}=\vec{\omega}\times\vec{P}$ and the cross-product of the vectors was defined as $$\vec{v}=\det \left[A\right];$$ matrix $A$ was defined as the cross-product matrix $$[A]=\left(\begin{matrix}\hat{x} &\hat{y} &\hat{z}\\0 &0 &{\theta}^{'}\\ \frac{L}{2}\cos\theta &\frac{L}{2}\sin\theta &0\end{matrix}\right).$$ Of course it works out, but my question is if I defined $$\vec{P}=\left(\begin{matrix}\frac{L}{2}\cos\theta \\ \frac{L}{2}\sin\theta\end{matrix}\right)$$ and rotational velocity as a tensor $\langle\omega\rangle$ such that $$\langle \omega\rangle= \left(\begin{matrix}0 &{-{\theta}}^{'}\\ {\theta}^{'} &0\end{matrix}\right),$$ then I could easily avoid using the third $z$ co-ordinate into the picture and also satisfy the linear map $$\vec{v}=\langle\omega\rangle\vec{P}.$$ Why then is $\omega$ forced to be a vector and not a tensor? Does it solve some purpose in physics or does it make the maths simple?

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    $\begingroup$ While one usually sees $\vec{\omega}=\dot{\theta} \hat{z}$ in introductory examples, this is not required and in more advanced examples it is not true. For example, the axis of rotation of a spinning top will generally not point straight up but will rather precess around the $z$-axis. So the cross product definition is more general than the tensor definition you give. $\endgroup$ – Semiclassical Nov 20 '16 at 23:53
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Gibbs Vector Algebra is the default introductory "Vector Algebra" of Physics since the early 1900's, having won out over Hamilton's Quaternions which were the 1st generally known 3D system to treat the Vector as a mathematical object of its own instead of doing everything in Cartesian coordinates.

Gibbs Vector algebra is encumbered by the often poorly made polar/axial vector distinction despite its near universal use and the prominent use of the axial vectors to represent angular dynamics quantities.

Tensors aren't usually introduced until late undergrad courses when not left to grad level courses altogether, and then only in fields that need the added generality.

There is an alternative that predates Tensors that is seeing some new popularity: Hestenes recovery, promotion, of "Geometric Algebra" (Gassmann and Clifford's own coinage, way prior to Cartan et al). Grassmann actually developed his "Extensive Algebra" at the same time Hamilton was creating the Quaternion Algebra. Grassmann's work, its significance as a true "Vector Algebra", was largely ignored for several decades until Clifford took up and built on Grassmann's ideas.

Clifford's early death left an opening for Gibbs Vector Algebra and its successful use by Heaviside in Electro-Magnetism to gain popularity in Physics despite Gibbs' own recognition of the need for "Multiple Algebras", his introduction of Dyadics only a few years after his initial Vector Algebra.

https://arxiv.org/pdf/1509.00501.pdf is a recent sumary of the history

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