This plot I made in desmos has the vectors $v_1=[9 \quad 0]^T$ and $v_2=[9\quad 1]^T$. If we connect the two vectors, then we can form a right triangle, where the right angle is at the point $(9,0)$ (or between $v_1$ and $v_1-v_2$). After rotating the two vectors, $v_1$ and $v_2$, $\alpha$ degrees (for a maximum effect rotate the animation by around 45 degrees) with a rotation matrix, it seems from the plot that there is not a right angle anymore between the rotated $v_1$ and rotated $v_1-v_2$. I checked this by using a sheet of paper and checking whether the angle was a 90 degree or angle or not. Furthermore, if the rotation is by 90 degrees, then the right triangle appears again over the y-axis.
I did the same in latex and the same phenomenon seems to occur there. Why is this? is there a problem with the programs? or is my intuition wrong? In my understanding, rotating both vectors is equivalent to rotating the axes, which would not break the right angle present in the original triangle. That's why I am very confused. Moreover, I am aware the rotation matrices should preserve angles as stated in this other question here. So, I don't really know what's going on with this construction that angles do not seem to be preserved.
Edit: I chose the value $\alpha=28^\circ$ and found a line perpendicular to $v_1$ rotated, which I found by using the negative of the inverse slope, i.e., $y=mx$, then $y=\dfrac{-x}{m}$ forms a line perpendicular to the original line.
The update plot is here.
I also upload a screenshot for reference. 
I cannot reproduce this effect (although visually it does appear to be true). I first tried using a protractor to measure the angle, but I got a right angle. Then I tried graphing the perpendicular line to $v_1$ by using the negative of the inverse slope. Since the slope is the tangent of the angle, my slope was the negative cotangent:
$$y=−\cot\left(\frac{a}{180} \pi\right)x + h$$
where $h$ is the rather-unimportant y-intercept.
But I found that when I calculated $h$ so that the tangent line and $v_1 - v_2$ intersected at the tip of $v_1$, the two lines were parallel.
(I believe the reason for this 'effect' is that it's harder to see right angles when they are rotated.)