2

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.

Math Guy
  • 359
  • In spherical coordinates, a vector's direction is captured by two angles $\theta$ and $\varphi$. You can extend this to $n$ dimensions, so that the interaction of $n-1$ angles determines the vector's direction. See here. – user170231 Mar 13 '20 at 15:13

2 Answers2

4

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.

Adam Rubinson
  • 24,300
  • That's one way of doing it anyway. You could instead define a position vector using n-dimensional Cartesian coordinates, but this definition would be only to do with lengths and nothing to do with angles. For example in 3 dimensions, (3,4,-7) = 3(1,0,0) + 4(0,1,0) + -7(0,0,1). It's the same thing with higher dimensions. This formulation is also widely used because the unit vectors form an orthonogonal basis (orthonormal actually) of the vector space $R^3$ ( or $R^n$ if you did it in the n-th dimension). See: https://en.wikipedia.org/wiki/Orthonormal_basis – Adam Rubinson Mar 13 '20 at 15:29
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.

Kraigolas
  • 1,569
Howardli621
  • 365