I am trying to understand the axis-angle/Rodrigues' rotation formula with https://en.wikipedia.org/w/index.php?title=Rodrigues%27_rotation_formula. As the article says, I can obtain my rotation axis $k$ by cross-product multiplying two intersecting vectors $a$ and $b$ that define a plane.
My confusion is that the so obtained $k$ is just a directional vector, not rooted at any particular point in the coordinate system. But to serve as a rotation axis, the vector has to be rooted at a certain point. Say, I compute $a\times b=k=(0, 0, 1)$. If I root the rotation axis at different starting points, the rotations are different, although the directional vector $k$ is still $(0, 0, 1)$.
So in a nutshell, I don't understand how the rotation axis is rooted instead of being merely directional.
The formulas shown in the Wikipedia article for Rodrigues' rotation formula are for rotation around an axis through the origin.
In order to rotate around some other axis, first perform a translation $T$ that maps some point on your desired rotation axis to the origin; then perform a rotation by the desired angle around the axis through the origin parallel to your desired rotation axis; then perform the inverse of the translation $T.$