# Quaternions

Eric Bainville - Mar 2007

This page is an introduction to quaternions, oriented towards their application to geometry. Properties and definitions used in other articles are given here in a condensed form. Interested readers should refer to the numerous articles found on the Internet on quaternions. Good starting points are the Wikipedia and MathWorld articles.

## Definitions

There are several different ways to define a quaternion. For applications to geometry, the most convenient definition is the following:

• A quaternion is a pair (s,u), where s is a real, and u a real 3-vector.
• Real x and quaternion (x,0) are identified, x is the real part of (x,u).
• 3-vector u and quaternion (0,u) are identified, u is the vector part of (x,u).
• (s,u) + (t,v) = (s + t,u + v), sum of two quaternions (commutative).
• (s,u) × (t,v) = (s.t - <u,v>,u×v + s.v + t.u), product of two quaternions (associative, but non-commutative), <u,v> is the usual dot product in R3, and u×v the usual cross product in R3.
• (s,u)* = (s,-u), conjugate of a quaternion.
• |(s,u)| = √(s2 + |u|2), norm of a quaternion.
• <(s,u),(t,v)> = s.t + <u,v>, dot product of two quaternions.

Sum and product are compatible with the identification with reals. Sum, product, and dot product are compatible with the identification with vectors (product of vector quaternions corresponds to cross product). The set Q of quaternions can be seen as a normed vector space of dimension 4, and a non-commutative R algebra. We have the following properties:

• q-1 = (1/|q|2) × q*, is the unique inverse of q (non zero).
• q** = q, conjugate of conjugate.
• (p + q)* = p* + q*, conjugate of sum.
• (p × q)* = q* × p*, conjugate of product.
• |q|2 = q × q*, link between conjugate and norm.
• The real part of q is (q + q*)/2.
• The vector part of q is (q - q*)/2.
• The dot product <p,q> is (p × q* + q × p*)/2.