Can three vectors have dot product less than $0$? 
Can three vectors in the $xy$ plane have $uv<0$ and $vw<0$ and $uw<0$?

If we take $u=(1,0)$ and $v=(-1,2)$ and $w=(-1,-2)$
$$uv=1\times(-1)=-1$$
$$uw=1\times(-1)+0\times(-2)=-1$$
$$vw=-1\times(-1)+2\times(-2)=-3$$
is there anyway to show that without examples, just working with $u=(u_1,u_2)$, $v=(v_1,v_2)$ and $w=(w_1,w_2)$, I mean analytically?

The dot product between two vectors is negative when the angle $\theta$ between them is greater than a right angle, since 
$$
u \cdot v = |u||v|\cos(\theta).
$$
So when three vectors point more or less to the vertices of an equilateral triangle all three dot products will be negative.
