Describing vectors: why do we need one angle in 2D, but three in 3D?

Question

When describing vectors in 2D, we need one angle. Why do we need three angles when describing vectors in 3D? I've tried thinking about this in depth, but I'm stumped. It seems like we'd need two and that's it.

Answer 1

You only need two to uniquely define the vector's direction, but if you want to define "roll" you need a third angle.

Answer 2

To describe the position of a point in 3D you need only two angles, but to describe the orientation of a 3D object requires three degrees of freedom.

But in 2D we only need one degree of freedom to describe orientation (rotation in the plane), so what links the 2D and 3D case? It is precisely the following:

Say that two planes (through the origin) $A, B$ are partially orthogonal if there are vectors $0\ne a \in A$ and $0 \ne b \in B$ such that $a$ and $b$ are orthogonal. In 2D, you can have at most one plane because there is only one; in 3D, you can have at most three pairwise partially-orthogonal (PPO) planes. This reasoning extends to 4D, where you can have six PPO planes, and so need six degrees of freedom to describe the orientation of a body. In general, in dimension $n$ there are at most $$ {n\choose 2} = \frac{n(n-1)}2 $$ PPO planes, and this is how many degrees of freedom you need to describe an orientation.

Answer 3

Given any two non-zero vectors, the cosine of the angle between them is defined as the direction cosine. In $n$-dimensional space there are $n$ mutually orthogonal coordinate vectors from the origin. Hence, given an arbitrary non-zero vector, there are $n$ direction cosines, one between each of the coordinate vectors and the given vector. It is easy to show that sum of the squares of these cosines is equal to $1$. For example, in two dimensions, $\,\cos(\theta_1)^2 + \cos(\theta_2)^2 = 1.\,$ The two angles are complementary to each other. However, the cosine is an even function so that given only $\,\cos(\theta_1)\,$ all we know is that $\,\cos(\theta_2) = \pm\sqrt{1-\cos(\theta_1)^2}.\,$ There are two possible values for $\,\cos(\theta_2)\,$ and for each of those two values of $\,\theta_2.\,$ The same thing happens in $n$ dimensions. Given any $\,n-1\,$ of the direction cosines, the remaining direction Despite this, in $n$ dimensions, the values of all of the $n$ direction cosines of a given vector determines the direction of the vector uniquely although there is a dependency between them.

A related construction is given by the use in $n$-dimensional space of spherical coordinates

in which the coordinates consist of a radial coordinate $r$, and $n-1$ angular coordinates and here only $n-1$ angles are needed to specify the direction of a non-zero vector.