Vectors and outer products

Let $\mathbf{a, \; b}$ be unitary vectors and perpendicular to each other. Let $\mathbf{u}$ be a vector defined as:

$$\mathbf{u}= \left ( \dots \left ( \left ( \left ( \mathbf{a}\times \mathbf{b} \right )\times \mathbf{b} \right )\times \mathbf{b} \right )\dots \right )\times \mathbf{b}$$

where $\mathbf{b}$ is repeated $2011$ times. Prove that:

$$\mathbf{u}=-\mathbf{a \times b}$$

Solution

Since $\mathbf{a, \; b}$ are unitary this means that $\mathbf{|a|=|b|=1}$. Also, since they are perpendicular this means that $\mathbf{ab=0}$ or $\sin (\widehat{\mathbf{a, b}})=1$.  

Therefore,

$$\mathbf{a\times b}=\left | a \right |\left | b \right |\sin \left ( \widehat{\mathbf{a, b}} \right )\mathbf{n}=\mathbf{n}$$

and $\mathbf{a\times b}$ is the unitary vector that is perpendicular to the plane of $\mathbf{a, b}$. In a similar manner we get that $\mathbf{\left ( a\times b \right )\times b}$ is the unitary and opposite vector of the vector $\mathbf{a}$, the vector $\mathbf{\left ( \left ( a\times b \right )\times b \right )\times b}$ is the unitary and opposite vector of the vector $\mathbf{a\times b}$. We also note that:

$$\mathbf{\left( \left ( \left ( a\times b \right )\times b \right )\times b\right) \times b=a}$$

Now, since $2011=4\cdot 502 + 3$ it follows that:

$$\begin{align*}
\mathbf{u} &=\left ( \dots \left ( \left ( \left ( \mathbf{a}\times \mathbf{b} \right )\times \mathbf{b} \right )\times \mathbf{b} \right )\dots \right )\times \mathbf{b} \\
 &=\mathbf{\left ( \left ( a\times b \right )\times b \right )\times b} \\
 &=\mathbf{-a \times b}
\end{align*}$$

proving the assertion.