What is wrong in my attempt to rotate two orthogonal vectors?

Question

Let $V=(1,0,0)$ and $W=(0,1,0)$ be two vector. Consider the following transformation:

$\pi/4$ rotation around the $z$-axis $V$ and $W$ concurrent counterclockwise, then $\pi/4$ rotating the result concurrent in direction from $y$-axis to $z$-axis (around $x$-axis) with fixed origin.

I think the result vectors is $V^\prime=(1,1,1)$ and $W^\prime=(-1,1,1)$ up to positive rescaling. and the above transformation is angle-preserving. but $\left<V^\prime,W^\prime\right>\neq0$!!

What is wrong in my attempt?

Answer 1

By putting your pieces of clues together and some guesswork I reach the conclusion that the second transformation is no rotation at all. It looks like it's just about adding $e_z$ to the vector (which is a shearing transformation).

The rotation preserves orthogonality all right even if you seem to be scaling the result as well. But the second does not, if $v$, $w$ and $e_z$ are orthogonal:

$$\langle v+e_z, w+e_z\rangle = \langle v,w\rangle + \langle v,e_z\rangle + \langle e_z, w\rangle + \langle e_z, e_z \rangle = \langle e_z, e_z\rangle = 1$$

This explains exactly how you get your result $\langle V', W'\rangle = 1\ne 0$

Answer 2

If the only transformations you perform are rotations around the origin, leaving the origin fixed, it is impossible to transform the pair of vectors $(1,0,0)$ and $(0,1,0)$ into the pair of vectors $(1,1,1)$ and $(-1,1,1).$

In fact, you start out with two vectors in the $x,y$ plane; after a rotation around the $z$ axis the resulting vectors are still in the $x,y$ plane. Then you rotate by angle $\frac\pi4$ around the $x$ axis, which should put the vectors in the plane $y=z.$

And indeed your two "result" vectors are in the plane $y=z,$ which is fine. But you also allowed yourself to "rescale" the vectors. My guess (only a guess since you steadfastly refuse to show your work) is that you rescaled twice, once on the first rotation to get $(1,1,0)$ and $(-1,1,0),$ and then again on the second rotation. But on the second rotation, while you allowed yourself to "scale" the $y$ and $z$ coordinates equally, you did not scale the $x$ coordinate by the same amount. Since you scaled by different amounts in different directions, you changed the angles of some pairs of vectors, including the pair you were trying to work with.