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

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.
