# Maths - Isometry Properties of 5d Multivectors

## Prerequisites

This page compares quaternion multiplication and orthogonal matrix multiplication as a means to represent rotation.

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

## Description

We want rotation and translation, these are reversible so we need a mathematical algebra that is also reversible, that is we need to always find the inverse, so:

if ,

multivector a translates b into c

then we need to be able to find the inverse a-1 which translates c into b

one condition which meets this requirement is:

a a†=scalar

So using the multiplication table here and multiplying out the terms of a a† gives:

e = +e*e+e1*e1+e2*e2+e3*e3+e4*e4+e5*e5+e12*e12+e13*e13+e14*e14
+e15*e15+e23*e23+e24*e24+e25*e25+e34*e34+e35*e35+e45*e45+e123*e123+e214*e214+e125*e125+e134*e134+e315*e315+e145*e145+e324*e324+e235*e235+e425*e425+-e12*e345+e1234*e1234+e1253*e1253+e1245*e1245+e3145*e3145+e2345*e2345+e12345*e12345

e1 = +e*e1+e1*e+e2*e12+e3*e13+e4*e14+e5*e15+e12*e2+e13*e3+e14*e4
+e15*e5+e23*e123-e24*e214+e25*e125+e34*e134-e35*e315+e45*e145+e123*e23-e214*e24+e125*e25+e134*e34-e315*e35+e145*e45-e324*e1234-e235*e1253-e425*e1245-e345*e3145-e1234*e324-e1253*e235-e1245*e425-e3145*e345+e2345*e12345+e12345*e2345

e2 = +e*e2-e1*e12+e2*e+e3*e23+e4*e24+e5*e25-e12*e1-e13*e123+e14*e214
-e15*e125+e23*e3+e24*e4+e25*e5-e34*e324+e35*e235-e45*e425-e123*e13+e214*e14-e125*e15-e134*e1234-e315*e1253-e145*e1245-e324*e34+e235*e35-e425*e45+e345*e2345-e1234*e134-e1253*e315-e1245*e145+e3145*e12345+e2345*e345+e12345*e3145

e3 = +e*e3-e1*e13-e2*e23+e3*e+e4*e34+e5*e35+e12*e123-e13*e1-e14*e134
+e15*e315-e23*e2+e24*e324-e25*e235+e34*e4+e35*e5+e45*e345+e123*e12-e214*e1234-e125*e1253-e134*e14+e315*e15+e145*e3145+e324*e24-e235*e25+e425*e2345+e345*e45-e1234*e214-e1253*e125+e1245*e12345+e3145*e145+e2345*e425+e12345*e1245

e4 = +e*e4-e1*e14-e2*e24-e3*e34+e4*e+e5*e45-e12*e214+e13*e134-e14*e1
-e15*e145-e23*e324-e24*e2+e25*e425-e34*e3-e35*e345+e45*e5-e123*e1234-e214*e12+e125*e1245+e134*e13+e315*e3145-e145*e15-e324*e23+e235*e2345+e425*e25-e345*e35-e1234*e123+e1253*e12345+e1245*e125+e3145*e315+e2345*e235+e12345*e1253

e5 = +e*e5-e1*e15-e2*e25-e3*e35-e4*e45+e5*e+e12*e125-e13*e315+e14*e145
-e15*e1+e23*e235-e24*e425-e25*e2+e34*e345-e35*e3-e45*e4+e123*e1253+e214*e1245+e125*e12+e134*e3145-e315*e13+e145*e14+e324*e2345+e235*e23-e425*e24+e345*e34+e1234*e12345+e1253*e123+e1245*e214+e3145*e134+e2345*e324+e12345*e1234

e12 = -e*e12+e1*e2-e2*e1-e3*e123+e4*e214-e5*e125+e12*e+e13*e23+e14*e24
+e15*e25-e23*e13-e24*e14-e25*e15-e34*e1234+e35*e1253-e45*e1245+e123*e3-e214*e4+e125*e5-e134*e324-e315*e235-e145*e425+e324*e134+e235*e315+e425*e145-e345*e12345+e1234*e34-e1253*e35+e1245*e45-e3145*e2345+e2345*e3145+e12345*e345

e13 = -e*e13+e1*e3+e2*e123-e3*e1-e4*e134+e5*e315-e12*e23+e13*e+e14*e34
+e15*e35+e23*e12+e24*e1234-e25*e1253-e34*e14-e35*e15+e45*e3145-e123*e2-e214*e324-e125*e235+e134*e4-e315*e5+e145*e345+e324*e214+e235*e125-e425*e12345-e345*e145-e1234*e24+e1253*e25-e1245*e2345-e3145*e45+e2345*e1245+e12345*e425

e14 = -e*e14+e1*e4-e2*e214+e3*e134-e4*e1-e5*e145-e12*e24-e13*e34+e14*e
+e15*e45-e23*e1234+e24*e12+e25*e1245+e34*e13-e35*e3145-e45*e15-e123*e324+e214*e2+e125*e425-e134*e3+e315*e345+e145*e5+e324*e123-e235*e12345-e425*e125-e345*e315+e1234*e23-e1253*e2345-e1245*e25+e3145*e35+e2345*e1253+e12345*e235

e15 = -e*e15+e1*e5+e2*e125-e3*e315+e4*e145-e5*e1-e12*e25-e13*e35-e14*e45
+e15*e+e23*e1253-e24*e1245+e25*e12+e34*e3145+e35*e13+e45*e14+e123*e235+e214*e425-e125*e2+e134*e345+e315*e3-e145*e4-e324*e12345-e235*e123-e425*e214-e345*e134-e1234*e2345-e1253*e23+e1245*e24-e3145*e34+e2345*e1234+e12345*e324

e23 = -e*e23-e1*e123+e2*e3-e3*e2+e4*e324-e5*e235+e12*e13-e13*e12-e14*e1234
+e15*e1253+e23*e+e24*e34+e25*e35-e34*e24-e35*e25-e45*e2345+e123*e1-e214*e134-e125*e315+e134*e214+e315*e125-e145*e12345-e324*e4+e235*e5-e425*e345+e345*e425+e1234*e14-e1253*e15-e1245*e3145+e3145*e1245+e2345*e45+e12345*e145

e24 = -e*e24+e1*e214+e2*e4-e3*e324-e4*e2+e5*e425+e12*e14+e13*e1234-e14*e12
-e15*e1245-e23*e34+e24*e+e25*e45+e34*e23+e35*e2345-e45*e25-e123*e134-e214*e1+e125*e145+e134*e123-e315*e12345-e145*e125+e324*e3-e235*e345-e425*e5+e345*e235-e1234*e13-e1253*e3145+e1245*e15+e3145*e1253-e2345*e35+e12345*e315

e25 = -e*e25-e1*e125+e2*e5+e3*e235-e4*e425-e5*e2+e12*e15-e13*e1253+e14*e1245
-e15*e12-e23*e35-e24*e45+e25*e-e34*e2345+e35*e23+e45*e24+e123*e315+e214*e145+e125*e1-e134*e12345-e315*e123-e145*e214-e324*e345-e235*e3+e425*e4+e345*e324-e1234*e3145+e1253*e13-e1245*e14+e3145*e1234+e2345*e34+e12345*e134

e34 = -e*e34-e1*e134+e2*e324+e3*e4-e4*e3-e5*e345-e12*e1234+e13*e14-e14*e13
+e15*e3145+e23*e24-e24*e23-e25*e2345+e34*e+e35*e45-e45*e35-e123*e214+e214*e123-e125*e12345+e134*e1-e315*e145+e145*e315-e324*e2-e235*e425+e425*e235+e345*e5+e1234*e12-e1253*e1245+e1245*e1253-e3145*e15+e2345*e25+e12345*e125

e35 = -e*e35+e1*e315-e2*e235+e3*e5+e4*e345-e5*e3+e12*e1253+e13*e15-e14*e3145
-e15*e13+e23*e25+e24*e2345-e25*e23-e34*e45+e35*e+e45*e34+e123*e125-e214*e12345-e125*e123-e134*e145-e315*e1+e145*e134-e324*e425+e235*e2+e425*e324-e345*e4-e1234*e1245-e1253*e12+e1245*e1234+e3145*e14-e2345*e24+e12345*e214

e45 = -e*e45-e1*e145+e2*e425-e3*e345+e4*e5-e5*e4-e12*e1245+e13*e3145+e14*e15
-e15*e14-e23*e2345+e24*e25-e25*e24+e34*e35-e35*e34+e45*e-e123*e12345-e214*e125+e125*e214-e134*e315+e315*e134+e145*e1-e324*e235+e235*e324-e425*e2+e345*e3-e1234*e1253+e1253*e1234+e1245*e12-e3145*e13+e2345*e23+e12345*e123

e123 = -e*e123-e1*e23+e2*e13-e3*e12-e4*e1234+e5*e1253+e12*e3-e13*e2+e14*e324
-e15*e235+e23*e1+e24*e134-e25*e315+e34*e214-e35*e125-e45*e12345+e123*e-e214*e34+e125*e35-e134*e24+e315*e25-e145*e2345-e324*e14+e235*e15+e425*e3145-e345*e1245+e1234*e4-e1253*e5+e1245*e345-e3145*e425+e2345*e145+e12345*e45

e214 = -e*e214+e1*e24-e2*e14-e3*e1234+e4*e12+e5*e1245-e12*e4+e13*e324+e14*e2
-e15*e425+e23*e134-e24*e1-e25*e145-e34*e123-e35*e12345+e45*e125+e123*e34+e214*e-e125*e45-e134*e23+e315*e2345+e145*e25-e324*e13-e235*e3145+e425*e15+e345*e1253+e1234*e3-e1253*e345-e1245*e5+e3145*e235-e2345*e315+e12345*e35

e125 = -e*e125-e1*e25+e2*e15-e3*e1253+e4*e1245-e5*e12+e12*e5+e13*e235-e14*e425
-e15*e2+e23*e315-e24*e145+e25*e1-e34*e12345+e35*e123-e45*e214-e123*e35+e214*e45+e125*e-e134*e2345-e315*e23+e145*e24+e324*e3145-e235*e13+e425*e14-e345*e1234+e1234*e345+e1253*e3-e1245*e4-e3145*e324+e2345*e134+e12345*e34

e134 = -e*e134-e1*e34-e2*e1234+e3*e14-e4*e13+e5*e3145+e12*e324+e13*e4-e14*e3
-e15*e345-e23*e214-e24*e123-e25*e12345+e34*e1+e35*e145+e45*e315+e123*e24+e214*e23-e125*e2345+e134*e-e315*e45-e145*e35-e324*e12+e235*e1245-e425*e1253+e345*e15+e1234*e2+e1253*e425-e1245*e235-e3145*e5+e2345*e125+e12345*e25

e315 = -e*e315+e1*e35-e2*e1253-e3*e15+e4*e3145+e5*e13+e12*e235-e13*e5-e14*e345
+e15*e3-e23*e125-e24*e12345+e25*e123+e34*e145-e35*e1-e45*e134-e123*e25+e214*e2345+e125*e23+e134*e45+e315*e-e145*e34-e324*e1245-e235*e12+e425*e1234+e345*e14-e1234*e425+e1253*e2+e1245*e324-e3145*e4-e2345*e214+e12345*e24

e145 = -e*e145-e1*e45-e2*e1245+e3*e3145+e4*e15-e5*e14+e12*e425-e13*e345+e14*e5
-e15*e4-e23*e12345+e24*e125+e25*e214-e34*e315-e35*e134+e45*e1-e123*e2345-e214*e25-e125*e24+e134*e35+e315*e34+e145*e+e324*e1253-e235*e1234-e425*e12+e345*e13+e1234*e235-e1253*e324+e1245*e2-e3145*e3+e2345*e123+e12345*e23

e324 = -e*e324-e1*e1234+e2*e34-e3*e24+e4*e23+e5*e2345-e12*e134-e13*e214-e14*e123
-e15*e12345-e23*e4+e24*e3+e25*e345-e34*e2+e35*e425+e45*e235+e123*e14+e214*e13+e125*e3145+e134*e12-e315*e1245+e145*e1253+e324*e-e235*e45-e425*e35-e345*e25+e1234*e1-e1253*e145+e1245*e315-e3145*e125-e2345*e5+e12345*e15

e235 = -e*e235-e1*e1253-e2*e35+e3*e25+e4*e2345-e5*e23-e12*e315-e13*e125-e14*e12345
+e15*e123+e23*e5+e24*e345-e25*e3+e34*e425+e35*e2-e45*e324-e123*e15-e214*e3145+e125*e13+e134*e1245+e315*e12-e145*e1234+e324*e45+e235*e-e425*e34-e345*e24+e1234*e145+e1253*e1-e1245*e134+e3145*e214-e2345*e4+e12345*e14

e425 = -e*e425-e1*e1245+e2*e45+e3*e2345-e4*e25+e5*e24-e12*e145-e13*e12345+e14*e125
+e15*e214+e23*e345-e24*e5+e25*e4-e34*e235-e35*e324-e45*e2+e123*e3145-e214*e15-e125*e14-e134*e1253+e315*e1234+e145*e12+e324*e35+e235*e34+e425*e-e345*e23-e1234*e315+e1253*e134+e1245*e1-e3145*e123-e2345*e3+e12345*e13

e345 = -e*e345-e1*e3145+e2*e2345-e3*e45+e4*e35-e5*e34-e12*e12345+e13*e145+e14*e315
+e15*e134-e23*e425-e24*e235-e25*e324+e34*e5-e35*e4+e45*e3-e123*e1245+e214*e1253-e125*e1234-e134*e15-e315*e14-e145*e13+e324*e25+e235*e24+e425*e23+e345*e+e1234*e125-e1253*e214+e1245*e123+e3145*e1-e2345*e2+e12345*e12

e1234 = +e*e1234+e1*e324+e2*e134+e3*e214+e4*e123+e5*e12345-e12*e34+e13*e24-e14*e23
-e15*e2345-e23*e14+e24*e13-e25*e3145-e34*e12-e35*e1245-e45*e1253+e123*e4+e214*e3-e125*e345+e134*e2+e315*e425-e145*e235+e324*e1-e235*e145+e425*e315-e345*e125+e1234*e-e1253*e45-e1245*e35-e3145*e25-e2345*e15+e12345*e5

e1253 = +e*e1253+e1*e235+e2*e315+e3*e125+e4*e12345-e5*e123+e12*e35-e13*e25-e14*e2345
+e15*e23+e23*e15-e24*e3145-e25*e13-e34*e1245+e35*e12+e45*e1234-e123*e5+e214*e345+e125*e3-e134*e425+e315*e2+e145*e324+e324*e145+e235*e1-e425*e134+e345*e214+e1234*e45+e1253*e-e1245*e34-e3145*e24-e2345*e14+e12345*e4

e1245 = +e*e1245+e1*e425+e2*e145+e3*e12345-e4*e125-e5*e214-e12*e45-e13*e2345+e14*e25
-e15*e24-e23*e3145-e24*e15+e25*e14+e34*e1253+e35*e1234-e45*e12-e123*e345-e214*e5-e125*e4+e134*e235-e315*e324+e145*e2-e324*e315+e235*e134+e425*e1-e345*e123+e1234*e35+e1253*e34+e1245*e-e3145*e23-e2345*e13+e12345*e3

e3145 = +e*e3145+e1*e345+e2*e12345-e3*e145-e4*e315-e5*e134-e12*e2345+e13*e45-e14*e35
+e15*e34+e23*e1245+e24*e1253+e25*e1234+e34*e15-e35*e14+e45*e13+e123*e425-e214*e235+e125*e324-e134*e5-e315*e4-e145*e3+e324*e125-e235*e214+e425*e123+e345*e1+e1234*e25+e1253*e24+e1245*e23+e3145*e-e2345*e12+e12345*e2

e2345 = +e*e2345+e1*e12345-e2*e345-e3*e425-e4*e235-e5*e324+e12*e3145+e13*e1245+e14*e1253
+e15*e1234-e23*e45+e24*e35-e25*e34-e34*e25+e35*e24-e45*e23-e123*e145+e214*e315-e125*e134-e134*e125+e315*e214-e145*e123-e324*e5-e235*e4-e425*e3-e345*e2+e1234*e15+e1253*e14+e1245*e13+e3145*e12+e2345*e+e12345*e1

e12345 = +e*e12345+e1*e2345+e2*e3145+e3*e1245+e4*e1253+e5*e1234-e12*e345-e13*e425-e14*e235
-e15*e324-e23*e145-e24*e315-e25*e134-e34*e125-e35*e214-e45*e123-e123*e45-e214*e35-e125*e34-e134*e25-e315*e24-e145*e23-e324*e15-e235*e14-e425*e13-e345*e12+e1234*e5+e1253*e4+e1245*e3+e3145*e2+e2345*e1+e12345*e

simplifying, by combining and canceling out terms, we get:

e = scalar = +e*e+e1*e1+e2*e2+e3*e3+e4*e4+e5*e5+e12*e12+e13*e13+e14*e14+e15*e15+e23*e23+e24*e24+e25*e25+e34*e34+e35*e35+e45*e45+e123*e123+e214*e214+e125*e125+e134*e134+e315*e315+e145*e145+e324*e324+e235*e235+e425*e425+e345*e345+e1234*e1234+e1253*e1253+e1245*e1245+e3145*e3145+e2345*e2345+e12345*e12345
0 = e1 = e*e1+e2*e12+e3*e13+e4*e14+e5*e15+e23*e123-e24*e214+e25*e125+e34*e134-e35*e315+e45*e145-e324*e1234-e235*e1253-e425*e1245-e345*e3145+e2345*e12345
0 = e2 = e*e2-e1*e12+e3*e23+e4*e24+e5*e25-e13*e123+e14*e214-e15*e125-e34*e324+e35*e235-e45*e425-e134*e1234-e315*e1253-e145*e1245+e345*e2345+e3145*e12345
0 = e3 = e*e3-e1*e13-e2*e23+e4*e34+e5*e35+e12*e123-e14*e134+e15*e315+e24*e324-e25*e235+e45*e345-e214*e1234-e125*e1253+e145*e3145+e425*e2345+e1245*e12345
0 = e4 = e*e4-e1*e14-e2*e24-e3*e34+e5*e45-e12*e214+e13*e134-e15*e145-e23*e324+e25*e425-e35*e345-e123*e1234+e125*e1245+e315*e3145+e235*e2345+e1253*e12345
0 = e5 = e*e5-e1*e15-e2*e25-e3*e35-e4*e45+e12*e125-e13*e315+e14*e145+e23*e235-e24*e425+e34*e345+e123*e1253+e214*e1245+e134*e3145+e324*e2345+e1234*e12345

0 = e1234 = e*e1234+e1*e324+e2*e134+e3*e214+e4*e123+e5*e12345-e12*e34+e13*e24-e14*e23-e15*e2345-e25*e3145-e35*e1245-e45*e1253-e125*e345+e315*e425-e145*e235
0 = e1253 = e*e1253+e1*e235+e2*e315+e3*e125+e4*e12345-e5*e123+e12*e35-e13*e25-e14*e2345+e15*e23-e24*e3145-e34*e1245+e45*e1234+e214*e345-e134*e425+e145*e324
0 = e1245 = e*e1245+e1*e425+e2*e145+e3*e12345-e4*e125-e5*e214-e12*e45-e13*e2345+e14*e25-e15*e24-e23*e3145+e34*e1253+e35*e1234-e123*e345+e134*e235-e315*e324
0 = e3145 = e*e3145+e1*e345+e2*e12345-e3*e145-e4*e315-e5*e134-e12*e2345+e13*e45-e14*e35+e15*e34+e23*e1245+e24*e1253+e25*e1234+e123*e425-e214*e235+e125*e324
0 = e2345 = e*e2345+e1*e12345-e2*e345-e3*e425-e4*e235-e5*e324+e12*e3145+e13*e1245+e14*e1253+e15*e1234-e23*e45+e24*e35-e25*e34-e123*e145+e214*e315-e125*e134

0 = e12345 = e*e12345+e1*e2345+e2*e3145+e3*e1245+e4*e1253+e5*e1234-e12*e345-e13*e425-e14*e235-e15*e324-e23*e145-e24*e315-e25*e134-e34*e125-e35*e214-e45*e123

all the other equations cancel out completely.

How do we solve these equations?

One possibility is to do the same as we did with the 3D case and let a*=a as follows:

e = e12345
e1 = e2345
e2 = e3145
e3 = e1245
e4 = e1253
e5 = e1234
e12 = -e345
e13 = -e425
e14 = -e235
e15 = -e324
e23 = -e145
e24 = -e315
e25 = -e134
e34 = -e125
e35 = -e214
e45 = -e123

then:

scalar = e = 2*(e*e+e1*e1+e2*e2+e3*e3+e4*e4+e5*e5+e12*e12+e13*e13+e14*e14+e15*e15+e23*e23+e24*e24+e25*e25+e34*e34+e35*e35+e45*e45)

0 = e*e1+e2*e12+e3*e13+e4*e14+e5*e15-e23*e45+e24*e35-e25*e34
0 = e*e2-e1*e12+e3*e23+e4*e24+e5*e25-e13*-e45+e14*-e35-e15*-e34
0 = e*e3-e1*e13-e2*e23+e4*e34+e5*e35+e12*-e45-e14*-e25+e15*-e24
e4 = e*e4-e1*e14-e2*e24-e3*e34+e5*e45-e12*-e35+e13*-e25-e15*-e23-e23*-e15+e25*-e13-e35*-e12+e45*e5-e34*e3-e24*e2-e14*e1+e4*e
e5 = e*e5-e1*e15-e2*e25-e3*e35-e4*e45+e12*-e34-e13*-e24+e14*-e23+e23*-e14-e24*-e13+e34*-e12-e45*e4-e35*e3-e25*e2-e15*e1+e5*e

-e1234 = +e*e5+e1*-e15+e2*-e25+e3*-e35+e4*-e45+e5*e-e12*e34+e13*e24-e14*e23-e15*e1-e25*e2-e35*e3-e45*e4+e34*-e12-e24*-e13+e23*-e14
-e1253 = +e*e4+e1*-e14+e2*-e24+e3*-e34+e4*e-e5*-e45+e12*e35-e13*e25-e14*e1+e15*e23-e24*e2-e34*e3+e45*e5-e35*-e12+e25*-e13-e23*-e15
e1245 = +e*e3+e1*-e13+e2*-e23+e3*e-e4*-e34-e5*-e35-e12*e45-e13*e1+e14*e25-e15*e24-e23*e2+e34*e4+e35*e5+e45*-e12-e25*-e14+e24*-e15
e3145 = +e*e2+e1*-e12+e2*e-e3*-e23-e4*-e24-e5*-e25-e12*e1+e13*e45-e14*e35+e15*e34+e23*e3+e24*e4+e25*e5-e45*-e13+e35*-e14-e34*-e15
e2345 = +e*e1+e1*e-e2*-e12-e3*-e13-e4*-e14-e5*-e15+e12*e2+e13*e3+e14*e4+e15*e5-e23*e45+e24*e35-e25*e34+e45*-e23-e35*-e24+e34*-e25

e12345 = e*e+e1*e1+e2*e2+e3*e3+e4*e4+e5*e5+e12*e12+e13*e13+e14*e14+e15*e15+e23*e23+e24*e24+e25*e25+e34*e34+e35*e35+e45*e45
So again the vector and the pseudovector are equal?

what is the problem???

lets try:

e = -e12345
e1 = e2345
e2 = e3145
e3 = e1245
e4 = e1253
e5 = e1234
e12 = e345
e13 = e425
e14 = e235
e15 = e324
e23 = e145
e24 = e315
e25 = e134
e34 = e125
e35 = e214
e45 = e123

this gives:

e = scalar = +e*e+e1*e1+e2*e2+e3*e3+e4*e4+e5*e5+e12*e12+e13*e13+e14*e14+e15*e15+e23*e23+e24*e24+e25*e25+e34*e34+e35*e35+e45*e45+e45*e45+e35*e35+e34*e34+e25*e25+e24*e24+e23*e23+e15*e15+e14*e14+e13*e13+e12*e12+e5*e5+e4*e4+e3*e3+e2*e2+e1*e1+e*e
0 = e1 = e*e1+e2*e12+e3*e13+e4*e14+e5*e15+e23*e45-e24*e35+e25*e34+e34*e25-e35*e24+e45*e23-e15*e5-e14*e4-e13*e3-e12*e2-e1*e
0 = e2 = e*e2-e1*e12+e3*e23+e4*e24+e5*e25-e13*e45+e14*e35-e15*e34-e34*e15+e35*e14-e45*e13-e25*e5-e24*e4-e23*e3+e12*e1-e2*e
0 = e3 = e*e3-e1*e13-e2*e23+e4*e34+e5*e35+e12*e45-e14*e25+e15*e24+e24*e15-e25*e14+e45*e12-e35*e5-e34*e4+e23*e2+e13*e1-e3*e
0 = e4 = e*e4-e1*e14-e2*e24-e3*e34+e5*e45-e12*e35+e13*e25-e15*e23-e23*e15+e25*e13-e35*e12-e45*e5+e34*e3+e24*e2+e14*e1-e4*e
0 = e5 = e*e5-e1*e15-e2*e25-e3*e35-e4*e45+e12*e34-e13*e24+e14*e23+e23*e14-e24*e13+e34*e12+e45*e4+e35*e3+e25*e2+e15*e1-e5*e

0 = e1234 = e*e5+e1*e15+e2*e25+e3*e35+e4*e45-e5*e-e12*e34+e13*e24-e14*e23-e15*e1-e25*e2-e35*e3-e45*e4-e34*e12+e24*e13-e23*e14
0 = e1253 = e*e4+e1*e14+e2*e24+e3*e34-e4*e-e5*e45+e12*e35-e13*e25-e14*e1+e15*e23-e24*e2-e34*e3+e45*e5+e35*e12-e25*e13+e23*e15
0 = e1245 = e*e3+e1*e13+e2*e23-e3*e-e4*e34-e5*e35-e12*e45-e13*e1+e14*e25-e15*e24-e23*e2+e34*e4+e35*e5-e45*e12+e25*e14-e24*e15
0 = e3145 = e*e2+e1*e12-e2*e-e3*e23-e4*e24-e5*e25-e12*e1+e13*e45-e14*e35+e15*e34+e23*e3+e24*e4+e25*e5+e45*e13-e35*e14+e34*e15
0 = e2345 = e*e1-e1*e-e2*e12-e3*e13-e4*e14-e5*e15+e12*e2+e13*e3+e14*e4+e15*e5-e23*e45+e24*e35-e25*e34-e45*e23+e35*e24-e34*e25

0 = e12345 = -e*e+e1*e1+e2*e2+e3*e3+e4*e4+e5*e5-e12*e12-e13*e13-e14*e14-e15*e15-e23*e23-e24*e24-e25*e25-e34*e125-e35*e35-e45*e45

cancelling out gives:

e = scalar = +e*e+e1*e1+e2*e2+e3*e3+e4*e4+e5*e5+e12*e12+e13*e13+e14*e14+e15*e15+e23*e23+e24*e24+e25*e25+e34*e34+e35*e35+e45*e45+e45*e45+e35*e35+e34*e34+e25*e25+e24*e24+e23*e23+e15*e15+e14*e14+e13*e13+e12*e12+e5*e5+e4*e4+e3*e3+e2*e2+e1*e1+e*e
0 = e1 = e23*e45-e24*e35+e25*e34+e34*e25-e35*e24+e45*e23
0 = e2 = -e13*e45+e14*e35-e15*e34-e34*e15+e35*e14-e45*e13
0 = e3 = +e12*e45-e14*e25+e15*e24+e24*e15-e25*e14+e45*e12
0 = e4 = -e12*e35+e13*e25-e15*e23-e23*e15+e25*e13-e35*e12
0 = e5 = e12*e34-e13*e24+e14*e23+e23*e14-e24*e13+e34*e12

0 = e1234 = -e12*e34+e13*e24-e14*e23-e34*e12+e24*e13-e23*e14
0 = e1253 = e12*e35-e13*e25+e15*e23+e35*e12-e25*e13+e23*e15
0 = e1245 = -e12*e45+e14*e25-e15*e24-e45*e12+e25*e14-e24*e15
0 = e3145 = e13*e45-e14*e35+e15*e34+e45*e13-e35*e14+e34*e15
0 = e2345 = -e23*e45+e24*e35-e25*e34-e45*e23+e35*e24-e34*e25

0 = e12345 = -e*e+e1*e1+e2*e2+e3*e3+e4*e4+e5*e5-e12*e12-e13*e13-e14*e14-e15*e15-e23*e23-e24*e24-e25*e25-e34*e34-e35*e35-e45*e45

Pure Rotation

If all ex, ey, ez and exyz input terms set to zero gives:

 e = a.e * b.e 0 0 0 - a.exy * b.exy - a.ezx * b.ezx - a.eyz * b.eyz 0 ex = 0 0 0 0 0 0 0 0 ey = 0 0 0 0 0 0 0 0 ez = 0 0 0 0 0 0 0 0 exy = a.exy * b.e 0 0 0 + a.e * b.exy + a.eyz * b.ezx - a.ezx * b.eyz 0 ezx = a.ezx * b.e 0 0 0 - a.eyz * b.exy + a.e * b.ezx + a.exy * b.eyz 0 eyz = a.eyz * b.e 0 0 0 + a.ezx * b.exy - a.exy * b.ezx + a.e * b.eyz 0 exyz = 0 0 0 0 0 0 0 0

If we include exyz back in we get:

 e = a.e * b.e 0 0 0 - a.exy * b.exy - a.ezx * b.ezx - a.eyz * b.eyz - a.exyz * b.exyz ex = 0 0 0 0 0 -0 - a.exyz * b.eyz - a.eyz * b.exyz ey = 0 0 0 0 0 - a.exyz * b.ezx 0 - a.ezx * b.exyz ez = 0 0 0 0 - a.exyz * b.exy 0 0 - a.exy * b.exyz exy = a.exy * b.e 0 0 0 + a.e * b.exy + a.eyz * b.ezx - a.ezx * b.eyz 0 ezx = a.ezx * b.e 0 0 0 - a.eyz * b.exy + a.e * b.ezx + a.exy * b.eyz 0 eyz = a.eyz * b.e 0 0 0 + a.ezx * b.exy - a.exy * b.ezx + a.e * b.eyz 0 exyz = a.exyz * b.e 0 0 0 0 0 0 + a.e * b.exyz

Pure Translation

If all exy, ezx and eyz input terms set to zero gives:

 e = a.e * b.e + a.ex * b.ex + a.ey * b.ey + a.ez * b.ez 0 0 0 - a.exyz * b.exyz ex = a.ex * b.e + a.e * b.ex 0 0 0 0 0 0 ey = a.ey * b.e 0 + a.e * b.ey 0 0 0 0 0 ez = a.ez * b.e 0 0 + a.e * b.ez 0 0 0 0 exy = 0 + a.ey * b.ex - a.ex * b.ey + a.exyz * b.ez 0 0 0 + a.ez * b.exyz ezx = 0 - a.ez * b.ex + a.exyz * b.ey + a.ex * b.ez 0 0 0 + a.ey * b.exyz eyz = 0 + a.exyz * b.ex + a.ez * b.ey - a.ey * b.ez 0 0 0 + a.ex * b.exyz exyz = a.exyz * b.e 0 0 0 0 0 0 + a.e * b.exyz

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

 Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (Fundamental Theories of Physics). This book is intended for mathematicians and physicists rather than programmers, it is very theoretical. It covers the algebra and calculus of multivectors of any dimension and is not specific to 3D modelling.