Can angles between 2 vectors be negative?

Question

Can angle between two be negative? Everywhere I read, it was written that angle is less then 180⁰ and greater than 0⁰. However, we can also rotate the other vector clockwise so we can have negative angle. So why is that not allowed?

Answer 1

1. A directed angle naturally can be negative:

   • as complex number arguments, the directed angles $356^\circ, -4^\circ$ and$-364^\circ$ are equivalent to one another;
   • however, $364^\circ$ is a greater angle of rotation than $4^\circ.$

2. On the other hand, an undirected angle is conventionally specified on some interval:

   • an angle in a triangle lies strictly between $0^\circ$ and $180^\circ;$ for example, the Sine Rule is formulated such that $30^\circ$ and $-30^\circ$ are not arbitrarily interchangeable;
   • the angle $\alpha$ between two vectors is defined on $[0^\circ,180^\circ];$ as such, it is nonstandard to refer to such an $89^\circ$ angle as $-89^\circ$ or $271^\circ$ or $-271^\circ$ or $449^\circ;$
   • the angle $\beta$ between two lines is defined on $[0^\circ,90^\circ].$

P.S. Each 'nonstandard' angle $x$ by can be converted to $\alpha,$ and likewise for $\beta:$ $$\alpha=\arccos(\cos(x))\\\beta=\left|\arctan(\tan(x))\right|.$$ For example, $$89^\circ=\arccos(\cos(-271^\circ))\\60^\circ=\left|\arctan(\tan(120^\circ))\right|.$$

Answer 2

In principle, there's nothing wrong with letting the angle be negative.

However, limiting the angle between $0^\circ$ and $180^\circ$ is natural, because it is the smallest of the two angles formed by a pair of non-parallel vectors in the plane they span, and it lends itself to the simple calculation of

$$\text{the angle between $x,y$} = \arccos \left( \frac{x \cdot y}{|x|\cdot |y|} \right)$$

The arccosine function has a range between $0$ and $\pi$ radians, meaning it naturally outputs the aforementioned smaller angle.

(In short, then, there is no issue, but this convention is both geometrically intuitive and computationally simple.)