TITLE: Does cross product have an identity?
QUESTION [14 upvotes]: Does cross product have an identity?  I.e. Does there exist some  $\vec{id}\in \mathbb{R}^3$ such that 
$$\vec{id} \times \vec{v} = \vec{v}\times \vec{id} = \vec{v}
$$ 
for all $\vec{v}\in \mathbb{R}^3$?

REPLY [6 votes]: Perhaps even easier.  Suppose such a vector $\vec{id}$ exists.  First note that $\vec{id} = 0$ does not work, so $\vec{id} \ne 0$.
Applying the desired property with $\vec{v} = \vec{id}$ we get
$$\vec{id} \times \vec{id} = \vec{id}$$
By antisymmetry, any vector cross itself is 0.  So this is a contradiction.