Norm of Quaternion

Question

Given a quaternion of the form,

$$q= a + bi + cj + dk$$

Which is the norm of $q$?

(1) $\sqrt{a^2+b^2+c^2+d^2}$

(2) $a^2+b^2+c^2+d^2$

This page from MathWorks says (1) but another page says (2). Wikipedia says (1). My lecture slides say:

$$|q|^2=a^2+b^2+c^2+d^2$$

but it is not clear whether $|q|$ is the norm or $|q|^2$ is the norm. It is not clear whether $||^2$ is just the L2 norm notation or it is the power of 2 notation.

I am trying to figure this out because I am trying to normalize and compute inverses of quaternions. Both of these operations depend on the norm.

Answer 1

From a practitioner point of view, you have two cases:

1. Your quaternion is not normalized, i.e., $Q Q^* \neq 1$, then the rotation is:

$$v' = Q v Q^{-1}$$

Where $Q^{-1} = Q^* / \|Q\|^2$ and $\|Q\|^2 = a^2 + b^2 + c^2 + d^2$.

As you see the nornalization term must be squared since Q appears twice in the quaternion ("sandwitch") product.

2. Your quaternion is unit quaternion, i.e., $Q Q^* = 1$, then the rotation is just:

$$v' = Q v Q^*$$

Now, a unit quaternion is just $Q / \|Q\|$ where $\|Q\| = \sqrt { a^2 + b^2 + c^2 + d^2}$.