# Is there an angle between vectors in n > 3 dimensions?

I can see why the dot product gives the angle between two vectors on $\mathbb {R}^{2}$, and that the angle between two vectors on $\mathbb {R}^{3}$ make sense, because you can take the plane defined by those two vectors, so it kind of falls back to $\mathbb {R}^{2}$, but what about $\mathbb {R}^{1}$ or $\mathbb {R}^{n}$ for $n \geq 4$? Is there such a thing as an angle in those other dimensions? I know there's such a thing as a distance, since by definition it's the length of a vector, but I can't grasp the concept of an angle then.

• In a word, yes. – Lord Shark the Unknown Jul 29 '18 at 7:56
• Maybe there are no angles in $n$ dimensions, but the dot product is still the product of the norms and the cosine of the angle ;-) – Yves Daoust Jul 29 '18 at 8:57

The notion of an angle exists in a general inner product space for example (beyond $\mathbb {R}^n$ and the dot product). In the case of dot product on two or three tuples, the angle concept coincides with the 'geometric' concept that we are first familier with, from school for example.