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?

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"?

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 $\vec{\sigma}$?