Physicists routinely wrote all 3 Pauli spin matrices as a vector. 


$$ \sigma\_1 = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0\end{array} \right) \hspace{0.25in}
\sigma\_2 = \left( \begin{array}{cc} 0 & -i \\ i & 0\end{array} \right)\hspace{0.25in}
\sigma\_3 = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1\end{array} \right) $$


This enables them to write shortcuts in quantum mechanics books such as:


$$ e^{ i\theta (\vec{v} \cdot \vec{\sigma})} = (\cos \theta) I + ( i \sin \theta) (\vec{v} \cdot \vec{\sigma}) $$


What kind of object is $\vec{\sigma}$ ? Is it a vector or a matrix or both?




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Another example is when we write the curl as a determinant.


$$ \nabla \times F = \left| \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F\_x & F\_y & F\_z \end{array} \right| $$


Curl is not the volume of any kind of box. Is there a rigorous geometric meaning for these "abuses of notation"?




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I asked this question on Math.SE and got a very literal answer, which did not help any. So I am asking here. You are welcome to answer there and call it a day.


Math.SE [What kind of object is the Pauli spin matrix $\vec{\sigma}$?](https://math.stackexchange.com/questions/1614858/what-kind-of-geometric-object-is-the-pauli-spin-matrix-vector-vec-sigma)



What kind of geometric object is the Pauli spin matrix vector $\vec{\sigma} = (\sigma_1, \sigma_2, \sigma_3)$?
The Pauli spin vector $\vec\sigma$ relates a two-component spinor $s={\alpha\choose \beta}$ to its corresponding three-component vector $\vec{v}=(a,b,c)$. In Dirac bra-ket notation the relation is written as
$$\vec{v}=\langle s|\vec\sigma|s\rangle,$$ 
in components $v\_i=s^\ast\sigma\_i s$. A pair of unitary operations $\pm U\in \text{SU}(2) $ on $s\mapsto s'$ is thereby mapped onto a rotation $R\in \text{SO}(3) $ of $\vec{v}\mapsto \vec{v}'=\langle s'|\vec\sigma|s'\rangle$. The mapping is illustrated by this [table](http://arxiv.org/abs/1312.3824)


![](https://ilorentz.org/beenakker/MO/spinor.png)


The usual vector identities do not apply to $\vec\sigma$, for example, instead of $\vec a\cdot(\vec b\times\vec a)=0$, one has $\vec \sigma\cdot(\vec v\times\vec \sigma)=-2i\,\vec v\cdot\vec\sigma$.


