Quaternions and the Cubic Formula

Introduction

Although not covered in the notes on Moodle, this topic may be of interest for those who want to know more about Quaternions. This short addendum introduces you to some higher level, complex, math relating to quaternions.

Cubic Formula

In algebra you are aware of the Quadratic Formula $a^2+b^2+c^2=0$ and the solution equation
$x=\frac{-b\pm{\sqrt{b^2-4ac}}}{2a}$

What is not taught is the solution to the Cubic Formula $ax^3+bx^2+cx+d=0$ . This formula is quite difficult to solve. One way to solve it was to use the simplified version of $x^2+px+q=0$ . This has the following solution:
$x=\xi\sqrt{\frac{-q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}+\xi^2\sqrt{\frac{-q}{2}-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}$

Where $\xi=1,\frac{-1+\sqrt{-3}}{2},\frac{-1-\sqrt{-3}}{2}$ .

Because complex numbers exist in a plane, multiplication of complex numbers yields a rotation in a 4D hypersphere.

4D Hypersphere

Uses the equation $a^2+b^2+c^2+d^2=1$ ; a 4D hypersphere has a 3D surface.

In geometric terms a circle is $s^1$ , a sphere is $s^2$ , and a hypersphere is $s^1$ .

Example

Given a point $(x,y,z)$ around an axis of a unit vector $[U_x,U_y,U_z]$ and an angle $\theta$ results in $(x',y',z')$ .

This expands to:
$x'i+y'j+z'k=q(xi+yj+zk)q^{-1}$ .

Where $q=cos\left ( \frac{\theta}{2} \right )+sin\left ( \frac{\theta}{2} \right )(U_xi+U_yj+U_zk)$ .

Note: this last equation has similarities to the Quaternion equation used in this course.

Exercises & Assignments

There are no exercises or assignments associated with this Addendum.

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