minimum value of vector expression If $\vec{a}$ and $\vec{b}$ are unit vectors and $\vec{c}$ satisfies  $$\vec{c}=-\vec{a}-(\vec{a} \cdot \vec{b})\vec{b} -\vec{a} \times\vec{b}$$ 
Then the minimum absolute value of  $(\vec{a} \times\vec{c} ) \cdot \vec{b}$ 
my attempt--
$(\vec{a} \times\vec{c} ) \cdot \vec{b}$ can be written down as  $\vec{a} \cdot(\vec{c}  \times \vec{b})$ and thus i calculated $\vec{c}  \times \vec{b}$ and then took the dot product with $\vec{a}$ but could not proceed to find out the minimum value.Please help me with this one.Thanks.

We start to simplify the expression for $(\vec{a} \times\vec{c} ) \cdot \vec{b}$ using properties of cross product:
$$(\vec{a} \times\vec{c} ) \cdot \vec{b}=$$
$$(\vec{a} \times (-\vec{a}-(\vec{a} \cdot \vec{b})\vec{b} -\vec{a} \times\vec{b}) ) \cdot \vec{b}=$$
$$(- \vec{a} \times \vec{a}-(\vec{a} \cdot \vec{b})(\vec{a} \times \vec{b}) -\vec{a}\times(\vec{a} \times\vec{b})) \cdot \vec{b}=$$
$$(-(\vec{a} \cdot \vec{b})(\vec{a} \times \vec{b}) - \vec{a}(\vec{a}\cdot \vec{b})+\vec{b}(\vec{a}\cdot \vec{a}) ) \cdot \vec{b}=$$
$$-(\vec{a} \cdot \vec{b})((\vec{a} \times \vec{b})\cdot \vec{b}) - (\vec{a}\cdot \vec{b})(\vec{a}\cdot \vec{b})+(\vec{b}\cdot \vec{b}) (\vec{a}\cdot \vec{a})=$$
$$(\vec{b}\cdot \vec{b}) (\vec{a}\cdot \vec{a}) - (\vec{a}\cdot \vec{b})(\vec{a}\cdot \vec{b}).$$
The last expression is the left hand side of Cauchy-Shwarz inequality for vectors $\vec{a}$ and $\vec{b}$, so it is at least zero and it is zero iff the vectors $\vec{a}$ and $\vec{b}$ are parallel, in case of their unit length iff $\vec{a}=\pm \vec{b}$.
