I've read that Givens rotations are linear transformations, but as we know linear transformations preserve the length of the vector transformed. However when I tried this with a 3 dimensional Givens rotation matrix G(1,2,45°) it seems that the length of the vector transformed changes, and when putting in it into a 3d grapher it the transformation resambles a plan projection.
Is that normal?
Welcome here Saad!
First point: linear transformations do not necessarily preserve the length. Take for example the map $A(x,y,z) = (3x,3y,3z)$: the length of the result is three times the original length!
Second point: luckily, rotations do preserve the length, so there should be something wrong with your matrix $G(1, 2, 45°) $ . How is it defined? If it is meant to be the rotation around the z-axis of 45°, it should be
$$ \begin{pmatrix} \sqrt{2}/2 & \sqrt{2}/2 & 0 \\ - \sqrt{2}/2 & \sqrt{2}/2 & 0 \\ 0 & 0 & 1 \end{pmatrix} $$