In one course a teacher said we were going to normalize a quaternion and did this:
$$\require{cancel} q = w + xi + yj + zk $$
$$ q` = \cancelto{1}{\frac{w}{w}} + \frac{x i}{w} + \frac{y j}{w} + \frac{z k}{w} $$
But on investigating on my own I found that a quaternion is normalized by dividing it by its norm. Is this another way to normalize a quaternion? And if not, does it have any other meaning?
"Normalizing" means rescaling in a way that is useful or meaningful in a given context. If you have an application or problem that requires you to rescale a quaternion to have real part $1$, then the way to do that is to divide by the real part!
(Unless it's a pure imaginary quaternion, then you're out of luck.)
The natural thing to do, however, is to rescale a quaternion so that it has norm $1$. Recall
$$ |w+x\mathbf{i}+y\mathbf{j}+z\mathbf{k}|=\sqrt{w^2+x^2+y^2+z^2}. $$
If you divide a quaternion $p$ by $|p|$ to get $u:=p/|p|$ has unit norm, i.e. $|u|=1$.
Can you see why $u$ must have norm $1$? In general, $|rp|=r|p|$ for nonnegative reals $r$, and even more generally still the norm is multiplicative, $|pq|=|p||q|$ for any two quaternions $p,q$.
