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I know vectors have both a magnitude and a direction, and I know that one may calculate the angle between two vectors.

I am reviewing an academic paper where one of the author has written " This is especially true when the angle between direction of phenomenon A and the direction of phenomenon B is low." This is in real coordinate space of 2 dimensions.

I don't want to be unnecessarily strict but this needs to be perfect, so I was wondering whether the vocabulary they use is correct or if it is an unacceptable shortcut.

ruadath
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Julian Wittische
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    Assuming the vectors we are talking about are in Euclidean space, $\mathbb R^n$, then a direction is just a unit vector. In that case, I don't see any problem with the author's language. If it is otherwise, please clarify. – Simon S Aug 04 '15 at 21:42
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    Given your edit adding "This is in real coordinate space of 2 dimensions", then the angle between two directions makes perfect sense. – Simon S Aug 04 '15 at 21:53
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    Agreed. It is just fine. – amcalde Aug 04 '15 at 21:55
  • I'd call such an angle "small" instead of "low". – Christian Blatter Aug 05 '15 at 18:56

2 Answers2

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Yes. One can define the angle between two nonzero vectors $\require{cancel}{\bf x}, {\bf y}$ to be the unique angle value $\theta \in [0, \pi]$ that satisfies $$\cos \theta = \frac{{\bf x} \cdot {\bf y}}{|{\bf x}| |{\bf y}|} .$$ Any other vectors pointing in the same direction as ${\bf x}, {\bf y}$ can be written as $\lambda {\bf x}, \mu {\bf y}$ for some $\lambda, \mu$, and the angle $\theta'$ between these new vectors satisfies $$\cos \theta' = \frac{\lambda {\bf x} \cdot \mu {\bf y}}{|\lambda {\bf x}| |\mu {\bf y}|} = \frac{\cancel{\lambda \mu} {\bf x} \cdot {\bf y}}{\cancel{\lambda \mu}|{\bf x}| |{\bf y}|} = \cos \theta.$$ So, the angle between two vectors defining directions doesn't depend on which particular vectors we choose, that is, the notion of angle between two directions is well defined.

If we are interested in working with directions rather than vectors, it is sometimes convenient to fix the above scaling of a vector, usually by insisting that the vectors representing a direction have length $1$; indeed, as Simon S points out, in this context such vectors (usually called unit vectors) are sometimes called direction vectors.

Travis Willse
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In $R^2$ a direction can be defined as a unique vector $(cosA,sinA)$. The angle between $(cosA_1,sinA_1)$ and $(cosA_2,sinA_2)$ is the unique $\theta$ in $[0,\pi]$ that satisfies $cos\theta=cos(A_1-A_2)$. Depending on the context,you might want a clarification of what the author means by a "low angle". How low is low?

DanielWainfleet
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