If the dot product between two vectors is $0$, are the two linearly independent? If we have vectors $V$ and $W$ in $\mathbb{R^n}$ and their dot product is $0$, are the two vectors linearly independent?
I can expand $V_1 \cdot V_2 = 0 \Rightarrow v_1w_1+...+v_nw_n = 0$, but I don't understand how this relates to linear independence.

More generally, Yes any orthogonal set of vectors are linearly independent. 
So if we consider $R^n$ together with the inner product being the dot product then if the set of vectors have dot product of zero then they are linearly indepedent:
Consider the set A = {$x_1$,$x_2$,...,$x_n$} 
Now we need to prove that the sets A is linearly indepedent set that is if you take a linear combinations of A then only solution that works is trivial one.
Consider $a_n \in R$
$a_1x_1 + ... + a_nx_n = 0$
do the dot product with $x_1$ notice everything will vanish since we have by assumption dot product is zero and you'll be left with 
$a_1x_1^{2} = 0$ so $a_1$ = 0 and you can proceed similarly and you'll find out that all constants are zero.
