Does rotation matrix keeps perpendicularity of two vectors?

I know my question is quite silly but let me explain my problem...
During the execution of my program, I wrote my own "lookAt" function wich takes into parameters two 3D points (eye position and view position) and a up vector. Everything is working great but, at one time, when I need to rotate my whole thing, I get a strange mistake I don't know how to solve :
I compute my two vectors (up vector and the view one (view - eye)), there I check this two vectors are perpendicular (v1.dot(v2) == 0). Then I multiply my two vectors by my rotation matrix and I get my two new vectors. But, when I check if the two vectors are perpendicular, v1.dot(v2) now equals to something like -x.xxxxxxxxxxxxxxxE-XX, XX like 16 or 17. I know the problem is coming from the precision of my numbers, but does someone knows how I can solve it to conserve the perpendicularity?
I really need this perpendicularity because I need to compute a new transform Matrix and with my problem, this one is not congruent, so I get an error.
Thanks!!

i did the same and had an error too, post you calculation code and i will try to help if i can

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