# Geomtric description of $(H_v)_{v\in X}$ for $X=\{(x,y)\in \mathbb{R}^2| x^2+y^2=1\}$ and $H_v=\{w\in \mathbb{R}^2|w\cdot v\geq 0\}$

Let $$X=\{(x,y)\in \mathbb{R}^2| x^2+y^2=1\}$$ the unit circle. For each vector $$v\in X$$ define $$H_v=\{w\in \mathbb{R}^2|w\cdot v\geq 0\}$$ where $$\cdot$$ is the dot product. How would someone geometrically describe the family of set $$(H_v)_{v\in X}$$?

$$H_v$$ is a half plane, bounded by the line passing through the origin and perpendicular to $$v$$. Of these two half-planes, $$H_v$$ is the one containing $$v$$.
$$H_v$$ is the set of vectors covering the half plane bounded by a line through (1,0) with the slope of the tangent to the unit circle at v.
• I guess the boundary line passes through the origin, since $v \cdot 0 \geq 0$, hence $0$ must be in $H_v$... – Hugo Sep 30 '18 at 22:29