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

Question

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.

Answer 1

In any dimension, any two vectors which are not collinear, span a plane. The angle between them is then exactly the same as the angle between two vectors in the plane.

Answer 2

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.

The same is true about the notion of perpendicular or orthogonal. The orthogonal then is not necessarilly the same as perpendicular in the usual geometric sense.