Will |v| = √(v · v) ever be negative? No, since length is always zero or positive.
Will it ever be zero? Yes, when v is the zero vector.
Another property of the dot product is:
(au + bv) · w = (au) · w + (bv) · w, where a and b are scalars
Here is the list of properties of the dot product:
(Trick Question: ) What is u · v · w ?