Multiplication of Two Quaternions

Introduction

Although not covered in the notes on Moodle, this topic may be of interest for those who want to know more about Quaternions. What is important in this topic is that $Q_{1}\times{Q_{2}}\neq{Q_{2}\times{Q_{1}}}$ . This is from the fundamental principle that rotations are not communative.

Multiplying Two Quaternions

Given $Q_{1}=(w_{1},V_{1})$ , where $V_{1}=(x_{1},y_{1},z_{1})$ and $Q_{2}=(w_{2},V_{2})$ where $V_{2}=(x_{2},y_{2},z_{2})$ , the multiplication of Quaternions follows:
$Q_{1}\times{Q_{2}}=(w_{1}w_{2}-V_{1}\cdot{V_{2}},w_{1}V_{2}+w_{2}V_{1}+V_{1}\times{V_{2}})$ .

Reversing the order of multiplication gives us:
$Q_{2}\times{Q_{1}}=(w_{2}w_{1}-V_{2}\cdot{V_{1}},w_{2}V_{1}+w_{1}V_{2}+V_{2}\times{V_{1}})$ .

The non-communative part of these two equations is $V_{1}\times{V_{2}}\neq{V_{2}\times{V_{1}}}$ (they are in opposite directions refer to the Vector Cross Product).

Sample Calculations

Here are the results of a sample calculation of multiplying two Quaternions:
quat_times_quat

Notice the opposite direction of the z-component of the results of the Cross Product.

Exercises & Assignments

There are no exercises or assignments associated with this Addendum.

Multiplication of Two Quaternions

Introduction

Multiplying Two Quaternions

Sample Calculations

Exercises & Assignments

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