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.

  • $\begingroup$ We rotate about an axis and we usually specify the axis with a vector. As in the 2D case, we can have clockwise or counter-clockwise rotations. If one vector is rotated about another, we can imagine the tip of the rotated vector tracing out an arc on a cone. $\endgroup$
    – John Douma
    May 23, 2022 at 16:59
  • $\begingroup$ If an intersecting line with an axis rotates, it generates a cone. When line is skew, it generates a hyperboloid of revolution of one sheet. $\endgroup$
    – Narasimham
    May 23, 2022 at 17:01

1 Answer 1


tl; dr:

Rotating vectors in three-space around a line

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).

  • $\begingroup$ I so badly want to understand how to use Epix but I can't seem to crack into the text pdf of it. Pensive $\endgroup$ May 29, 2022 at 20:41
  • $\begingroup$ @Aplateofmomos Do you mean the manual is hard to read, that the sample files are hard to use, something else...? $\endgroup$ May 29, 2022 at 22:44
  • $\begingroup$ The manual is hard to read but it may be that I don't have experience reading such manuals $\endgroup$ May 29, 2022 at 22:46
  • $\begingroup$ @Aplateofmomos Looking at and experimenting with the sample files is probably an easier way to get started. Have you got the software installed and running? $\endgroup$ May 30, 2022 at 2:14

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