how $3i \times 3i = 9i \times i$? (i is the unit vector and $\times$ is cross product) $i$ is the unit vector; didn't know how to write it.
I'm reading a text and somewhere it uses something like $ai \times bi = (ab)i \times i$ (implicitly). I can see why this is true geometrically, but the text says "by distributive law". Is there something I'm missing?
 A: Is the $\times$ means the cross product?
In usual vector operation, we have two kinds of operations, (inner, outer(or so called cross) products). 
For basis(kinds of unit vector) $\mathbb{e}_i$, the scalar product just changes its scale of vector. 
but its vector product(i mean inner, cross product) changes its direction(inner products gives the scalar, and cross products gives the vector). 
$i.e$  (assume we are in $R^3$)
For $\vec{A}=a_i\mathbb{e}_i$ and $\vec{B}=b_i\mathbb{e}_i$
For inner product between them 
$ \vec{A}\cdot \vec{B} =a_ib_j(\mathbb{e}_i \cdot \mathbb{e}_j)=a_ib_j \delta_{ij} =a_i b_i$
and for the cross product 
$ \vec{A} \times \vec{B} =a_i b_j (\mathbb{e}_i \times \mathbb{e}_j)=\epsilon_{ijk}a_ib_j\mathbb{e}_k$
Note the operation for vectors only applied for vectors, and operation for scalar only applied for scalars. 
If you know furthermore, please look for a any vector calculus algebra textbook, or you can see the explain on wikipedia (i.e http://en.wikipedia.org/wiki/Dot_product in section Properties, http://en.wikipedia.org/wiki/Cross_product in section Computing the cross product) 
