What is vector division? My question is: 
We have addition, subtraction and muliplication of vectors. Why cannot we define vector division? What is division of vectors?

Introduction. There are very good answers and mine will only supplement them. About "inverse" of dot product, imagine you have $\vec a \cdot \vec b=c$ and you know $\vec b$ and $c$ and want to know what is $\vec a$. What this problem means? It means that you have parallelogram of known area witch lies on known vector and you need to determine other vector. It turns out that there are many such $\vec a$, so you have a set of vectors, more precisely a curve. Every curve can be expressed as vector function. Now we will try to find this vector function.
Derivations.

*

*$\vec a \cdot \vec b=c$

*$(a_x i+a_y j )\cdot (b_x i+b_y j)=c$

*Let us call $a_x=x$ and $a_y=y$

*$(x i+y j )\cdot (b_x i+b_y j)=c$

*$x b_x+y b_y=c$

*$y=\frac{c- x b_x}{b_y}$

*$b_x= \vec b \cdot i$ and $b_y=\vec b \cdot j$

*$y=\frac{c- x \vec b \cdot i}{\vec b \cdot j}$

*Vector function of this function is:
$\vec a=x i+(\frac{c- x \vec b \cdot i}{\vec b \cdot j})j$

*It is equivalent to:
$\vec a = x i+\left (\frac{c- x |b|cos(\alpha)}{|b| sin(\alpha)} \right)j$, where $\alpha=\angle(\vec a;\vec b)$
Summery. These derivations can be carried out in both directions so:
$\vec a \cdot \vec b=c$ is equivalent to $\vec a=x i+(\frac{c- x \vec b \cdot i}{\vec b \cdot j})j$ and it is equivalent to $\vec a = x i+\left (\frac{c- x |b|cos(\alpha)}{|b| sin(\alpha)} \right)j$, where $\alpha=\angle(\vec a;\vec b)$
Similar answer.
You can find similar answer about finding "inverse" of cross product here:
Finding $\vec b$ from $\vec a \times \vec b$ , $\vec a$ and $\alpha=\angle(\vec a;\vec b)$ using only vector algebra..
