## Prerequisites

If you are not familiar with this subject you may like to look at the following pages first:

## using Multivectors

If we specify a velocity using a 3D multivector (explained here) and choose to define this as follows:

Scalar (e) = 1

Vector (ex,ey,ez) = vx,vy,vz

BiVector (eyz,ezx,exy) = 0

TriVector (exyz) = 1

where: | other definitions | ||

symbol |
description |
type |
units |

ex,ey,ez | basis vectors | vector | |

exyz | trivector | ||

vx,vy,vz | components of velocity in x,y and z dimensions | vector |

So the velocity multivector is:

v = 1 + ex * vx + ey * vy + ez * vz + exyz

We can new calculate the square of the velocity using multi-vector multiplication which gives:

v^{2} = x^{2} + y^{2} + z^{2} + ex * 2 * vx
+ ey * 2 * vy + ez * 2 * vz + eyz * 2 * vx + ezx * 2 * vy + exy * 2 * vz + exyz
* 2

We can new calculate m * v^{2}/2 which gives the following:

m * v^{2}/2 = (x^{2} + y^{2} + z^{2})*m/2+
ex * m * vx + ey * m * vy + ez * m * vz + eyz * m * vx + ezx * m * vy + exy
* m * vz + exyz * m

This has the following components:

Scalar (e) = (x^{2} + y^{2} + z^{2})*m/2 = kinetic
energy

Vector (ex,ey,ez) = m * vx ,m * vy, m * vz = linear momentum

BiVector (eyz,ezx,exy) = m * vx ,m * vy, m * vz = linear momentum

TriVector (exyz) = m = mass

So it is made up of constants for linear motion, for example in a perfectly elastic collision between two objects, if we calculate this quantity for both objects before the collision and after the collision, the total of both objects before the collision should be the same as the total after the collision.

### Adding rotational energy

The above only includes linear motion, it would be good if we could find a way to include rotational energy and momentum. So here we try adding a rotational velocity (w) to the velocity multivector as follows:

Scalar (e) = 1

Vector (ex,ey,ez) = vx,vy,vz

BiVector (eyz,ezx,exy) = wx,wy,wz

TriVector (exyz) = 1

where: | |||

symbol |
description |
type |
units |

w | angular velocity | bivector | s^{-1} |

angle | angle in radians | scalar | none |

t | time | scalar | s |

d ... /dt | rate of change |

where:

- ex,ey,ez = basis vectors
- exyz = represents tri-vector
- vx,vy,vz = components of velocity in x,y and z dimensions
- wx,wy,wz = components of angular velocity in x,y and z dimensions

So the velocity multivector is:

v = 1 + ex * vx + ey * vy + ez * vz +eyz * wx + ezx * wy + exy * wz + exyz

We can new calculate the square of this velocity which gives: