A Few Questions Concerning Vectors In my textbook, they provide a theorem to calculate the angle between two vectors:
$\cos\theta = \Large\frac{\vec{u} \cdot \vec{v}}{\|\vec{u}\|\|\vec{v}\|}$
My questions are, why does the angle have to be $0 \le \theta \le \pi$; and why do the vectors have to be in standard position?
Also, on the next page, the author writes, "the zero vector is orthogonal to every vector
because $0 \cdot \vec{u} = 0$;" why is that so?

Given two points $x$ and $y$ on the unit sphere $S^{n-1}\subset{\mathbb R}^n$ the spherical distance between them is the length of the shortest arc on $S^{n-1}$ connecting $x$ and $y$. The shortest arc obviously lies in the plane spanned by $x$ and $y$, and drawing a figure of this plane one sees that the length $\phi$ of the arc in question can be computed by means of the scalar product as $$\phi =\arccos(x\cdot y)\ \ \in[0,\pi]\ .$$ This length is then also called the angle between $x$ and $y$. 
When $u$ and $v$ are arbitrary nonzero vectors in ${\mathbb R}^n$ then $u':={u\over |u|}$ and $v':={v\over |v|}$ lie on $S^{n-1}$. Geometrical intuition tells us that $\angle(u,v)=\angle(u',v')$. Therefore one defines the angle $\phi\in[0,\pi]$ between $u$ and $v$ as
$$\phi:=\arccos{u\cdot v\over|u|\>|v|}\ .$$
