on the unit normal vector....

“When I use a word,’ Humpty Dumpty said in rather a scornful tone, ‘it means just what I choose it to mean — neither more nor less.’

’The question is,’ said Alice, ‘whether you can make words mean so many different things.’ Through the Looking Glass, by Lewis Carroll, page 367

So let me try to correct the record.  It is true that for a parametric curve r(t), the velocity vector v(t)=r'(t) is simply related to the unit tangent vector T(t)=r'(t)/||r'(t)||=v(t)/||v(t)||.  And it's also true that the acceleration vector is a(t)=v'(t)=r"(t).  While grading the exam I saw about fifty wishful but incorrect statements to the effect that there is a simple relationship between the unit normal vector and the acceleration:  N(t)=a(t)/||a(t)|| BUT THIS IS INCORRECT.

Like Humpty Dumpty, we can define things to mean whatever we want, and a(t)/||a(t)|| is a perfectly good unit vector. The people who thought of the unit normal vector as something useful were looking for more. They wanted N(t) to point in the direction that v(t) is turning toward AND to be orthogonal to v(t).  Unfortunately for the sake of simplicity a(t) can't do this job.  This is easy to see: sometimes a(t) points in the same direction that v does, for instance when v is increasing in magnitude but doesn't change direction.  In order to get what we need, we need to define a different unit vector N(t)=T'(t)/||T'(t)||, which is always perpendicular to the direction of motion.