How to visualise 3D vector rotation around a line?

Question

If we want to rotate a 2D vector we need only angle $\theta$ by which we want to rotate the vector. And we have only two possibilities: one for clockwise rotation and other for counterclockwise rotation. Pretty simple!

But things get complicated when we want to rotate 3D vectors. Mere giving an angle is not sufficient as we would have infinite possibilities for rotated vector. In fact, it is not hard to see that all possibilities of rotated vectors would make a cone, right?

So in 3D rotations we rotate a vector around a line, right?

But my problem is that I am not able to visualise how we are rotating a vector around a line. I don't need any rotation matrix or any other algebraic stuff. I just want to visualise the rotation through 3D diagrams.

It would be a great help if you spare some time to answer this question in detail.

You might simply take a 3D vector as an example, and show how we are getting a rotated vector around a line, for example z-axis.

Thank you.

Answer 1

tl; dr:

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In Euclidean geometry, two planes are orthogonal if every vector of one plane is orthogonal to every vector in the other. In four-space, for example, the planes $\{(u, v, 0, 0) : \text{$u$, $v$ real}\}$ and $\{(0, 0, x, y) : \text{$x$, $y$ real}\}$ are orthogonal.

In Euclidean $n$-space, a rotation acts on some collection of mutually-orthogonal planes and fixes whatever space is orthogonal to the direct sum. This gives Euclidean rotations in odd-dimensional spaces a slightly different character than rotations in even-dimensional spaces: Every rotation of an odd-dimensional space has at least one line that is fixed, but there exist rotations of even-dimensional spaces that fix only one point (the center of rotation).

In three-space there do not exist two orthogonal planes; a rotation acts on some plane (here the plane containing the tail of the red arrow) and fixes the orthogonal line, the axis (the line containing the black arrow).