Definition of direction of a 3D vector

Question

What is the formal definition of the direction of a 3D vector? In the plane, it can be defined as the (obtuse) angle formed by the position vector and $(1,0)$. In 3D space, it is the angles formed by the position vector and the three vectors in the canonical basis, that is, (1,0,0),(0,1,0) and (0,0,1)?

How about in the n-dimensional space? is there any formal definition of direction a vector?

Thanks in advance.

Answer 1

You're asking a similar question to here:

Analogue of spherical coordinates in $n$-dimensions

For two dimensions, you can use polar coordinates:

For 3 dimensions, you can use spherical coordinates:

For n dimensions, you can use hyperspherical coordinates.

But basically, in any n-dimensional space, you'll have one length coordinate and (n-1) angle coordinates.

Answer 2

In fact, you need to use 2 independent angles to precisely describe the direction of a 3D vector. This is easier to imagine in spherical coordinates:

$x=r\sin(\phi) \cos (\theta)$

$y=r\sin(\phi) \sin (\theta)$

$z=r\cos(\phi)$

You can think of $\theta$ as a parameter that controls the longitude while $\phi$ is a parameter that controls the latitude. Note that we only care about the direction of a vector, so $r$, the length of the vector is neglected.

In general, the direction of a vector in $\mathbb{R}^n$ needs $n-1$ independent angles to describe, but it is difficult to visualize in the higher dimension.