Computing dot products of linear combinations of unit vectors. 
For any unit vectors $v$ and $w$, find the dot products (actual numbers) of:
a) $v$ and $-v$
b) $v+w$ and $v-w$
c) $v-2w$ and $v+2w$

I have worked part a :
a) $(v) \cdot (-v) = \cos (180) = -1$
Not getting any ideas on how to work part b and c... Any help ?

Use the fact that dot products distribute over addition and dot products are commutative and scalars can be pulled out from either vector. For any scalar $c$, we have that:
\begin{align*}
(\vec v + c \vec w) \cdot (\vec v - c \vec w)
&= \vec v \cdot \vec v - c(\vec v \cdot \vec w) +  c(\vec v \cdot \vec w) - c^2(\vec w \cdot \vec w) \\
&= |\vec v|^2 - c^2|\vec w|^2 \\
\end{align*}
