# Finding the Axis of Rotation

I was given the following information: There is a frame $B$ which is rotated relative to frame $A$ and they share the same origin. The following vectors are in frame $A$: $(2, 1, 2)$ and $(-1, -2, 2)$. These two vectors also happen to lay on the $x$ axis and $z$ axis of Frame $B$ respectively.

I am tasked with finding the axis of rotation, and I also need to find the angle at which the frame $B$ was rotated in relation to frame $A$.

Through my knowledge I was able to find that the angle at which the frame was rotated was approximately 48.2 degrees, but I am unsure if this is correct and am unable to find the axis of rotation. I feel like I am missing something critical yet simple here.

• Finding the axis is equivalent to finding the fixed points of the rotation. Solve Rx=x. Commented Oct 6, 2015 at 22:54
• Very closely related: math.stackexchange.com/questions/1125203/… and math.stackexchange.com/questions/624348/… Commented Oct 6, 2015 at 22:59
• Note that if you multiply each vector by the scalar $\frac13$, the resulting vectors are two basis vectors from an orthonormal basis. Commented Oct 6, 2015 at 23:03

Because they line up with the new axes the vectors $(2,1,2)$ and $(-1,-2,2)$ will (once normalized) be the first and third rows of your rotation matrix respectively. Now you need to find the $y$-axis.
Once you have the matrix you just need to find the vector space that doesn't move. This is just solving the equation $Mx=x$ or rather $(M-I)x=0$.