The arithmatic on this page asumes the operands are a general case of a 5D multivector:
first multivector
+ a.e e + a.e1 e1 + a.e2 e2 + a.e3 e3 + a.e4 e4 + a.e5 e5 + a.e12 e12 + a.e13 e13 + a.e14 e14 + a.e15 e15 + a.e23 e23 + a.e24 e24 + a.e25 e25 + a.e34 e34 + a.e35 e35 + a.e45 e45 + a.e123 e123 + a.e214 e214 + a.e125 e125 + a.e134 e134 + a.e315 e315 + a.e145 e145 + a.e324 e324 + a.e235 e235 + a.e425 e425 + a.e345 e345 + a.e1234 e1234 + a.e1253 e1253 + a.e1245 e1245 + a.e3145 e3145 + a.e2345 e2345 + a.e12345 e12345
and a second multivector:
b.e e + b.e1 e1 + b.e2 e2 + b.e3 e3 + b.e4 e4 + b.e5 e5 + b.e12 e12 + b.e13 e13 + b.e14 e14 + b.e15 e15 + b.e23 e23 + b.e24 e24 + b.e25 e25 + b.e34 e34 + b.e35 e35 + b.e45 e45 + b.e123 e123 + b.e214 e214 + b.e125 e125 + b.e134 e134 + b.e315 e315 + b.e145 e145 + b.e324 e324 + b.e235 e235 + b.e425 e425 + b.e345 e345 + b.e1234 e1234 + b.e1253 e1253 + b.e1245 e1245 + b.e3145 e3145 + b.e2345 e2345 + b.e12345 e12345
Adding multi vectors numbers
Just add each component independently as follows:
e = a.e + b.e
e1 = a.e1 + b.e1
e2 = a.e2 + b.e2
e3 = a.e3 + b.e3
e4 = a.e4 + b.e4
e5 = a.e5 + b.e5
e12 = a.e12 + b.e12
e13 = a.e13 + b.e13
e14 = a.e14 + b.e14
e15 = a.e15 + b.e15
e23 = a.e23 + b.e23
e24 = a.e24 + b.e24
e25 = a.e25 + b.e25
e34 = a.e34 + b.e34
e35 = a.e35 + b.e35
e45 = a.e45 + b.e45
e123 = a.e123 + b.e123
e214 = a.e214 + b.e214
e125 = a.e125 + b.e125
e134 = a.e134 + b.e134
e315 = a.e315 + b.e315
e145 = a.e145 + b.e145
e324 = a.e324 + b.e324
e235 = a.e235 + b.e235
e425 = a.e425 + b.e425
e345 = a.e345 + b.e345
e1234 = a.e1234 + b.e1234
e1253 = a.e1253 + b.e1253
e1245 = a.e1245 + b.e1245
e3145 = a.e3145 + b.e3145
e2345 = a.e2345 + b.e2345
e12345 = a.e12345 + b.e12345
This operation will be coded in the multi5d class (see
this class here).
When adding blades of different grade then we cant reduce it further and we leave the + in the number.
For example:
3 + 4 e1 + 5 e12
added to
5 + 4 e2 + 3 e12
gives
8 + 4 e1 + 4 e2 + 8 e12
Subtracting multi vectors numbers
Just subtract each component independently as follows:
e = a.e - b.e
e1 = a.e1 - b.e1
e2 = a.e2 - b.e2
e3 = a.e3 - b.e3
e4 = a.e4 - b.e4
e5 = a.e5 - b.e5
e12 = a.e12 - b.e12
e13 = a.e13 - b.e13
e14 = a.e14 - b.e14
e15 = a.e15 - b.e15
e23 = a.e23 - b.e23
e24 = a.e24 - b.e24
e25 = a.e25 - b.e25
e34 = a.e34 - b.e34
e35 = a.e35 - b.e35
e45 = a.e45 - b.e45
e123 = a.e123 - b.e123
e214 = a.e214 - b.e214
e125 = a.e125 - b.e125
e134 = a.e134 - b.e134
e315 = a.e315 - b.e315
e145 = a.e145 - b.e145
e324 = a.e324 - b.e324
e235 = a.e235 - b.e235
e425 = a.e425 - b.e425
e345 = a.e345 - b.e345
e1234 = a.e1234 - b.e1234
e1253 = a.e1253 - b.e1253
e1245 = a.e1245 - b.e1245
e3145 = a.e3145 - b.e3145
e2345 = a.e2345 - b.e2345
e12345 = a.e12345 - b.e12345
This operation will be coded in the multi5d class (see
this class here).
Multiplying 5D multivectors numbers (geometric product)
The main type of multiplication, which is described here, is geometric multiplication. Each term can be calculated using simple rules as described here.
In order to make sure we work out all possible combinations of products, I suggest using a table. The entries in the table only shows the type and sign change of the product, it does not show its absolute value. We therefore need to prefix the product by its numerical value which is the real number which is the product of the numbers at the top and left headings.
So we can start by entering the above results in the table, the value to the left of the * is represented by the columns and the value to the right of the * is represented by the rows. So when calculating a^b the column headings are denoted by a.?? and the rows are denoted by b.??
So the finished table is:
| e | e1 | e2 | e3 | e4 | e5 | e12 | e13 | e14 | e15 | e23 | e24 | e25 | e34 | e35 | e45 | e123 | e214 | e125 | e134 | e315 | e145 | e324 | e235 | e425 | e345 | e1234 | e1253 | e1245 | e3145 | e2345 | e12345 |
| e1 | e | e12 | e13 | e14 | e15 | e2 | e3 | e4 | e5 | e123 | -e214 | e125 | e134 | -e315 | e145 | e23 | -e24 | e25 | e34 | -e35 | e45 | -e1234 | -e1253 | -e1245 | -e3145 | -e324 | -e235 | -e425 | -e345 | e12345 | e2345 |
| e2 | -e12 | e | e23 | e24 | e25 | -e1 | -e123 | e214 | -e125 | e3 | e4 | e5 | -e324 | e235 | -e425 | -e13 | e14 | -e15 | -e1234 | -e1253 | -e1245 | -e34 | e35 | -e45 | e2345 | -e134 | -e315 | -e145 | e12345 | e345 | e3145 |
| e3 | -e13 | -e23 | e | e34 | e35 | e123 | -e1 | -e134 | e315 | -e2 | e324 | -e235 | e4 | e5 | e345 | e12 | -e1234 | -e1253 | -e14 | e15 | e3145 | e24 | -e25 | e2345 | e45 | -e214 | -e125 | e12345 | e145 | e425 | e1245 |
| e4 | -e14 | -e24 | -e34 | e | e45 | -e214 | e134 | -e1 | -e145 | -e324 | -e2 | e425 | -e3 | -e345 | e5 | -e1234 | -e12 | e1245 | e13 | e3145 | -e15 | -e23 | e2345 | e25 | -e35 | -e123 | e12345 | e125 | e315 | e235 | e1253 |
| e5 | -e15 | -e25 | -e35 | -e45 | e | e125 | -e315 | e145 | -e1 | e235 | -e425 | -e2 | e345 | -e3 | -e4 | e1253 | e1245 | e12 | e3145 | -e13 | e14 | e2345 | e23 | -e24 | e34 | e12345 | e123 | e214 | e134 | e324 | e1234 |
| e12 | -e2 | e1 | e123 | -e214 | e125 | -e | -e23 | -e24 | -e25 | e13 | e14 | e15 | e1234 | -e1253 | e1245 | -e3 | e4 | -e5 | e324 | e235 | e425 | -e134 | -e315 | -e145 | e12345 | -e34 | e35 | -e45 | e2345 | -e3145 | -e345 |
| e13 | -e3 | -e123 | e1 | e134 | -e315 | e23 | -e | -e34 | -e35 | -e12 | -e1234 | e1253 | e14 | e15 | -e3145 | e2 | e324 | e235 | -e4 | e5 | -e345 | -e214 | -e125 | e12345 | e145 | e24 | -e25 | e2345 | e45 | -e1245 | -e425 |
| e14 | -e4 | e214 | -e134 | e1 | e145 | e24 | e34 | -e | -e45 | e1234 | -e12 | -e1245 | -e13 | e3145 | e15 | e324 | -e2 | -e425 | e3 | -e345 | -e5 | -e123 | e12345 | e125 | e315 | -e23 | e2345 | e25 | -e35 | -e1253 | -e235 |
| e15 | -e5 | -e125 | e315 | -e145 | e1 | e25 | e35 | e45 | -e | -e1253 | e1245 | -e12 | -e3145 | -e13 | -e14 | -e235 | -e425 | e2 | -e345 | -e3 | e4 | e12345 | e123 | e214 | e134 | e2345 | e23 | -e24 | e34 | -e1234 | -e324 |
| e23 | e123 | -e3 | e2 | -e324 | e235 | -e13 | e12 | e1234 | -e1253 | -e | -e34 | -e35 | e24 | e25 | e2345 | -e1 | e134 | e315 | -e214 | -e125 | e12345 | e4 | -e5 | e345 | -e425 | -e14 | e15 | e3145 | -e1245 | -e45 | -e145 |
| e24 | -e214 | -e4 | e324 | e2 | -e425 | -e14 | -e1234 | e12 | e1245 | e34 | -e | -e45 | -e23 | -e2345 | e25 | e134 | e1 | -e145 | -e123 | e12345 | e125 | -e3 | e345 | e5 | -e235 | e13 | e3145 | -e15 | -e1253 | e35 | -e315 |
| e25 | e125 | -e5 | -e235 | e425 | e2 | -e15 | e1253 | -e1245 | e12 | e35 | e45 | -e | e2345 | -e23 | -e24 | -e315 | -e145 | -e1 | e12345 | e123 | e214 | e345 | e3 | -e4 | -e324 | e3145 | -e13 | e14 | -e1234 | -e34 | -e134 |
| e34 | e134 | -e324 | -e4 | e3 | e345 | e1234 | -e14 | e13 | -e3145 | -e24 | e23 | e2345 | -e | -e45 | e35 | e214 | -e123 | e12345 | -e1 | e145 | -e315 | e2 | e425 | -e235 | -e5 | -e12 | e1245 | -e1253 | e15 | -e25 | -e125 |
| e35 | -e315 | e235 | -e5 | -e345 | e3 | -e1253 | -e15 | e3145 | e13 | -e25 | -e2345 | e23 | e45 | -e | -e34 | -e125 | e12345 | e123 | e145 | e1 | -e134 | e425 | -e2 | -e324 | e4 | e1245 | e12 | -e1234 | -e14 | e24 | -e214 |
| e45 | e145 | -e425 | e345 | -e5 | e4 | e1245 | -e3145 | -e15 | e14 | e2345 | -e25 | e24 | -e35 | e34 | -e | e12345 | e125 | -e214 | e315 | -e134 | -e1 | e235 | -e324 | e2 | -e3 | e1253 | -e1234 | -e12 | e13 | -e23 | -e123 |
| e123 | e23 | -e13 | e12 | e1234 | -e1253 | -e3 | e2 | -e324 | e235 | -e1 | -e134 | e315 | -e214 | e125 | e12345 | -e | e34 | -e35 | e24 | -e25 | e2345 | e14 | -e15 | -e3145 | e1245 | -e4 | e5 | -e345 | e425 | -e145 | -e45 |
| e214 | -e24 | e14 | e1234 | -e12 | -e1245 | e4 | -e324 | -e2 | e425 | -e134 | e1 | e145 | e123 | e12345 | -e125 | -e34 | -e | e45 | e23 | -e2345 | -e25 | e13 | e3145 | -e15 | -e1253 | -e3 | e345 | e5 | -e235 | e315 | -e35 |
| e125 | e25 | -e15 | e1253 | -e1245 | e12 | -e5 | -e235 | e425 | e2 | -e315 | e145 | -e1 | e12345 | -e123 | e214 | e35 | -e45 | -e | e2345 | e23 | -e24 | -e3145 | e13 | -e14 | e1234 | -e345 | -e3 | e4 | e324 | -e134 | -e34 |
| e134 | e34 | e1234 | -e14 | e13 | -e3145 | -e324 | -e4 | e3 | e345 | e214 | e123 | e12345 | -e1 | -e145 | -e315 | -e24 | -e23 | e2345 | -e | e45 | e35 | e12 | -e1245 | e1253 | -e15 | -e2 | -e425 | e235 | e5 | -e125 | -e25 |
| e315 | -e35 | e1253 | e15 | -e3145 | -e13 | -e235 | e5 | e345 | -e3 | e125 | e12345 | -e123 | -e145 | e1 | e134 | e25 | -e2345 | -e23 | -e45 | -e | e34 | e1245 | e12 | -e1234 | -e14 | e425 | -e2 | -e324 | e4 | e214 | -e24 |
| e145 | e45 | e1245 | -e3145 | -e15 | e14 | -e425 | e345 | -e5 | e4 | e12345 | -e125 | -e214 | e315 | e134 | -e1 | e2345 | e25 | e24 | -e35 | -e34 | -e | -e1253 | e1234 | e12 | -e13 | -e235 | e324 | -e2 | e3 | -e123 | -e23 |
| e324 | e1234 | -e34 | e24 | -e23 | -e2345 | e134 | e214 | e123 | e12345 | e4 | -e3 | -e345 | e2 | -e425 | -e235 | -e14 | -e13 | -e3145 | -e12 | e1245 | -e1253 | -e | e45 | e35 | e25 | -e1 | e145 | -e315 | e125 | e5 | -e15 |
| e235 | e1253 | e35 | -e25 | -e2345 | e23 | e315 | e125 | e12345 | -e123 | -e5 | -e345 | e3 | -e425 | -e2 | e324 | e15 | e3145 | -e13 | -e1245 | -e12 | e1234 | -e45 | -e | e34 | e24 | -e145 | -e1 | e134 | -e214 | e4 | -e14 |
| e425 | e1245 | -e45 | -e2345 | e25 | -e24 | e145 | e12345 | -e125 | -e214 | -e345 | e5 | -e4 | e235 | e324 | e2 | -e3145 | e15 | e14 | e1253 | -e1234 | -e12 | -e35 | -e34 | -e | e23 | e315 | -e134 | -e1 | e123 | e3 | -e13 |
| e345 | e3145 | -e2345 | e45 | -e35 | e34 | e12345 | -e145 | -e315 | -e134 | e425 | e235 | e324 | -e5 | e4 | -e3 | e1245 | -e1253 | e1234 | e15 | e14 | e13 | -e25 | -e24 | -e23 | -e | -e125 | e214 | -e123 | -e1 | e2 | -e12 |
| e1234 | e324 | e134 | e214 | e123 | e12345 | -e34 | e24 | -e23 | -e2345 | -e14 | e13 | -e3145 | -e12 | -e1245 | -e1253 | e4 | e3 | -e345 | e2 | e425 | -e235 | e1 | -e145 | e315 | -e125 | e | -e45 | -e35 | -e25 | -e15 | e5 |
| e1253 | e235 | e315 | e125 | e12345 | -e123 | e35 | -e25 | -e2345 | e23 | e15 | -e3145 | -e13 | -e1245 | e12 | e1234 | -e5 | e345 | e3 | -e425 | e2 | e324 | e145 | e1 | -e134 | e214 | e45 | e | -e34 | -e24 | -e14 | e4 |
| e1245 | e425 | e145 | e12345 | -e125 | -e214 | -e45 | -e2345 | e25 | -e24 | -e3145 | -e15 | e14 | e1253 | e1234 | -e12 | -e345 | -e5 | -e4 | e235 | -e324 | e2 | -e315 | e134 | e1 | -e123 | e35 | e34 | e | -e23 | -e13 | e3 |
| e3145 | e345 | e12345 | -e145 | -e315 | -e134 | -e2345 | e45 | -e35 | e34 | e1245 | e1253 | e1234 | e15 | -e14 | e13 | e425 | -e235 | e324 | -e5 | -e4 | -e3 | e125 | -e214 | e123 | e1 | e25 | e24 | e23 | e | -e12 | e2 |
| e2345 | e12345 | -e345 | -e425 | -e235 | -e324 | e3145 | e1245 | e1253 | e1234 | -e45 | e35 | -e34 | -e25 | e24 | -e23 | -e145 | e315 | -e134 | -e125 | e214 | -e123 | -e5 | -e4 | -e3 | -e2 | e15 | e14 | e13 | e12 | e | e1 |
| e12345 | e2345 | e3145 | e1245 | e1253 | e1234 | -e345 | -e425 | -e235 | -e324 | -e145 | -e315 | -e134 | -e125 | -e214 | -e123 | -e45 | -e35 | -e34 | -e25 | -e24 | -e23 | -e15 | -e14 | -e13 | -e12 | e5 | e4 | e3 | e2 | e1 | e |
In the above table we can see that some entries are commutative and some are anti-commutative, that is, if we swap rows and columns (or reflect in leading diagonal) some values remain the same and the others have their sign changed. We can also see that (A * B)† = B†* A† because if we rotate the whole table by 90 degrees then the entry will become its reversal.
I guess what we really need to know is that, given a multivector with numerical values: a.e, a.e1, a.e2, a.e3, a.e12, a.e31, a.e23 and a.e123 multiplied by a second multivector with numerical values: b.e, b.e1, b.e2, b.e3, b.e12, b.e31, b.e23 and b.e123 then what are the resulting numerical values. Multiplying out each term gives the following result:
e = +a.e*b.e+a.e1*b.e1+a.e2*b.e2+a.e3*b.e3+a.e4*b.e4+a.e5*b.e5-a.e12*b.e12-a.e13*b.e13-a.e14*b.e14
-a.e15*b.e15-a.e23*b.e23-a.e24*b.e24-a.e25*b.e25-a.e34*b.e34-a.e35*b.e35-a.e45*b.e45-a.e123*b.e123-a.e214*b.e214-a.e125*b.e125-a.e134*b.e134-a.e315*b.e315-a.e145*b.e145-a.e324*b.e324-a.e235*b.e235-a.e425*b.e425-a.e345*b.e345+a.e1234*b.e1234+a.e1253*b.e1253+a.e1245*b.e1245+a.e3145*b.e3145+a.e2345*b.e2345+a.e12345*b.e12345
e1 = +a.e*b.e1+a.e1*b.e-a.e2*b.e12-a.e3*b.e13-a.e4*b.e14-a.e5*b.e15+a.e12*b.e2+a.e13*b.e3+a.e14*b.e4
+a.e15*b.e5-a.e23*b.e123+a.e24*b.e214-a.e25*b.e125-a.e34*b.e134+a.e35*b.e315-a.e45*b.e145-a.e123*b.e23+a.e214*b.e24-a.e125*b.e25-a.e134*b.e34+a.e315*b.e35-a.e145*b.e45-a.e324*b.e1234-a.e235*b.e1253-a.e425*b.e1245-a.e345*b.e3145+a.e1234*b.e324+a.e1253*b.e235+a.e1245*b.e425+a.e3145*b.e345+a.e2345*b.e12345+a.e12345*b.e2345
e2 = +a.e*b.e2+a.e1*b.e12+a.e2*b.e-a.e3*b.e23-a.e4*b.e24-a.e5*b.e25-a.e12*b.e1+a.e13*b.e123-a.e14*b.e214
+a.e15*b.e125+a.e23*b.e3+a.e24*b.e4+a.e25*b.e5+a.e34*b.e324-a.e35*b.e235+a.e45*b.e425+a.e123*b.e13-a.e214*b.e14+a.e125*b.e15-a.e134*b.e1234-a.e315*b.e1253-a.e145*b.e1245+a.e324*b.e34-a.e235*b.e35+a.e425*b.e45+a.e345*b.e2345+a.e1234*b.e134+a.e1253*b.e315+a.e1245*b.e145+a.e3145*b.e12345-a.e2345*b.e345+a.e12345*b.e3145
e3 = +a.e*b.e3+a.e1*b.e13+a.e2*b.e23+a.e3*b.e-a.e4*b.e34-a.e5*b.e35-a.e12*b.e123-a.e13*b.e1+a.e14*b.e134
-a.e15*b.e315-a.e23*b.e2-a.e24*b.e324+a.e25*b.e235+a.e34*b.e4+a.e35*b.e5-a.e45*b.e345-a.e123*b.e12-a.e214*b.e1234-a.e125*b.e1253+a.e134*b.e14-a.e315*b.e15+a.e145*b.e3145-a.e324*b.e24+a.e235*b.e25+a.e425*b.e2345-a.e345*b.e45+a.e1234*b.e214+a.e1253*b.e125+a.e1245*b.e12345-a.e3145*b.e145-a.e2345*b.e425+a.e12345*b.e1245
e4 = +a.e*b.e4+a.e1*b.e14+a.e2*b.e24+a.e3*b.e34+a.e4*b.e-a.e5*b.e45+a.e12*b.e214-a.e13*b.e134-a.e14*b.e1
+a.e15*b.e145+a.e23*b.e324-a.e24*b.e2-a.e25*b.e425-a.e34*b.e3+a.e35*b.e345+a.e45*b.e5-a.e123*b.e1234+a.e214*b.e12+a.e125*b.e1245-a.e134*b.e13+a.e315*b.e3145+a.e145*b.e15+a.e324*b.e23+a.e235*b.e2345-a.e425*b.e25+a.e345*b.e35+a.e1234*b.e123+a.e1253*b.e12345-a.e1245*b.e125-a.e3145*b.e315-a.e2345*b.e235+a.e12345*b.e1253
e5 = +a.e*b.e5+a.e1*b.e15+a.e2*b.e25+a.e3*b.e35+a.e4*b.e45+a.e5*b.e-a.e12*b.e125+a.e13*b.e315-a.e14*b.e145
-a.e15*b.e1-a.e23*b.e235+a.e24*b.e425-a.e25*b.e2-a.e34*b.e345-a.e35*b.e3-a.e45*b.e4+a.e123*b.e1253+a.e214*b.e1245-a.e125*b.e12+a.e134*b.e3145+a.e315*b.e13-a.e145*b.e14+a.e324*b.e2345-a.e235*b.e23+a.e425*b.e24-a.e345*b.e34+a.e1234*b.e12345-a.e1253*b.e123-a.e1245*b.e214-a.e3145*b.e134-a.e2345*b.e324+a.e12345*b.e1234
e12 = +a.e*b.e12+a.e1*b.e2-a.e2*b.e1+a.e3*b.e123-a.e4*b.e214+a.e5*b.e125+a.e12*b.e-a.e13*b.e23-a.e14*b.e24
-a.e15*b.e25+a.e23*b.e13+a.e24*b.e14+a.e25*b.e15-a.e34*b.e1234+a.e35*b.e1253-a.e45*b.e1245+a.e123*b.e3-a.e214*b.e4+a.e125*b.e5+a.e134*b.e324+a.e315*b.e235+a.e145*b.e425-a.e324*b.e134-a.e235*b.e315-a.e425*b.e145-a.e345*b.e12345-a.e1234*b.e34+a.e1253*b.e35-a.e1245*b.e45-a.e3145*b.e2345+a.e2345*b.e3145-a.e12345*b.e345
e13 = +a.e*b.e13+a.e1*b.e3-a.e2*b.e123-a.e3*b.e1+a.e4*b.e134-a.e5*b.e315+a.e12*b.e23+a.e13*b.e-a.e14*b.e34
-a.e15*b.e35-a.e23*b.e12+a.e24*b.e1234-a.e25*b.e1253+a.e34*b.e14+a.e35*b.e15+a.e45*b.e3145-a.e123*b.e2+a.e214*b.e324+a.e125*b.e235+a.e134*b.e4-a.e315*b.e5-a.e145*b.e345-a.e324*b.e214-a.e235*b.e125-a.e425*b.e12345+a.e345*b.e145+a.e1234*b.e24-a.e1253*b.e25-a.e1245*b.e2345+a.e3145*b.e45+a.e2345*b.e1245-a.e12345*b.e425
e14 = +a.e*b.e14+a.e1*b.e4+a.e2*b.e214-a.e3*b.e134-a.e4*b.e1+a.e5*b.e145+a.e12*b.e24+a.e13*b.e34+a.e14*b.e
-a.e15*b.e45-a.e23*b.e1234-a.e24*b.e12+a.e25*b.e1245-a.e34*b.e13-a.e35*b.e3145+a.e45*b.e15+a.e123*b.e324+a.e214*b.e2-a.e125*b.e425-a.e134*b.e3-a.e315*b.e345+a.e145*b.e5-a.e324*b.e123-a.e235*b.e12345+a.e425*b.e125+a.e345*b.e315-a.e1234*b.e23-a.e1253*b.e2345+a.e1245*b.e25-a.e3145*b.e35+a.e2345*b.e1253-a.e12345*b.e235
e15 = +a.e*b.e15+a.e1*b.e5-a.e2*b.e125+a.e3*b.e315-a.e4*b.e145-a.e5*b.e1+a.e12*b.e25+a.e13*b.e35+a.e14*b.e45
+a.e15*b.e+a.e23*b.e1253-a.e24*b.e1245-a.e25*b.e12+a.e34*b.e3145-a.e35*b.e13-a.e45*b.e14-a.e123*b.e235-a.e214*b.e425-a.e125*b.e2-a.e134*b.e345+a.e315*b.e3-a.e145*b.e4-a.e324*b.e12345+a.e235*b.e123+a.e425*b.e214+a.e345*b.e134-a.e1234*b.e2345+a.e1253*b.e23-a.e1245*b.e24+a.e3145*b.e34+a.e2345*b.e1234-a.e12345*b.e324
e23 = +a.e*b.e23+a.e1*b.e123+a.e2*b.e3-a.e3*b.e2-a.e4*b.e324+a.e5*b.e235-a.e12*b.e13+a.e13*b.e12-a.e14*b.e1234
+a.e15*b.e1253+a.e23*b.e-a.e24*b.e34-a.e25*b.e35+a.e34*b.e24+a.e35*b.e25-a.e45*b.e2345+a.e123*b.e1+a.e214*b.e134+a.e125*b.e315-a.e134*b.e214-a.e315*b.e125-a.e145*b.e12345-a.e324*b.e4+a.e235*b.e5+a.e425*b.e345-a.e345*b.e425-a.e1234*b.e14+a.e1253*b.e15-a.e1245*b.e3145+a.e3145*b.e1245-a.e2345*b.e45-a.e12345*b.e145
e24 = +a.e*b.e24-a.e1*b.e214+a.e2*b.e4+a.e3*b.e324-a.e4*b.e2-a.e5*b.e425-a.e12*b.e14+a.e13*b.e1234+a.e14*b.e12
-a.e15*b.e1245+a.e23*b.e34+a.e24*b.e-a.e25*b.e45-a.e34*b.e23+a.e35*b.e2345+a.e45*b.e25+a.e123*b.e134-a.e214*b.e1-a.e125*b.e145-a.e134*b.e123-a.e315*b.e12345+a.e145*b.e125+a.e324*b.e3+a.e235*b.e345-a.e425*b.e5-a.e345*b.e235+a.e1234*b.e13-a.e1253*b.e3145-a.e1245*b.e15+a.e3145*b.e1253+a.e2345*b.e35-a.e12345*b.e315
e25 = +a.e*b.e25+a.e1*b.e125+a.e2*b.e5-a.e3*b.e235+a.e4*b.e425-a.e5*b.e2-a.e12*b.e15-a.e13*b.e1253+a.e14*b.e1245
+a.e15*b.e12+a.e23*b.e35+a.e24*b.e45+a.e25*b.e-a.e34*b.e2345-a.e35*b.e23-a.e45*b.e24-a.e123*b.e315-a.e214*b.e145+a.e125*b.e1-a.e134*b.e12345+a.e315*b.e123+a.e145*b.e214+a.e324*b.e345-a.e235*b.e3+a.e425*b.e4-a.e345*b.e324-a.e1234*b.e3145-a.e1253*b.e13+a.e1245*b.e14+a.e3145*b.e1234-a.e2345*b.e34-a.e12345*b.e134
e34 = +a.e*b.e34+a.e1*b.e134-a.e2*b.e324+a.e3*b.e4-a.e4*b.e3+a.e5*b.e345-a.e12*b.e1234-a.e13*b.e14+a.e14*b.e13
+a.e15*b.e3145-a.e23*b.e24+a.e24*b.e23-a.e25*b.e2345+a.e34*b.e-a.e35*b.e45+a.e45*b.e35+a.e123*b.e214-a.e214*b.e123-a.e125*b.e12345+a.e134*b.e1+a.e315*b.e145-a.e145*b.e315-a.e324*b.e2+a.e235*b.e425-a.e425*b.e235+a.e345*b.e5-a.e1234*b.e12-a.e1253*b.e1245+a.e1245*b.e1253+a.e3145*b.e15-a.e2345*b.e25-a.e12345*b.e125
e35 = +a.e*b.e35-a.e1*b.e315+a.e2*b.e235+a.e3*b.e5-a.e4*b.e345-a.e5*b.e3+a.e12*b.e1253-a.e13*b.e15-a.e14*b.e3145
+a.e15*b.e13-a.e23*b.e25+a.e24*b.e2345+a.e25*b.e23+a.e34*b.e45+a.e35*b.e-a.e45*b.e34-a.e123*b.e125-a.e214*b.e12345+a.e125*b.e123+a.e134*b.e145-a.e315*b.e1-a.e145*b.e134+a.e324*b.e425+a.e235*b.e2-a.e425*b.e324-a.e345*b.e4-a.e1234*b.e1245+a.e1253*b.e12+a.e1245*b.e1234-a.e3145*b.e14+a.e2345*b.e24-a.e12345*b.e214
e45 = +a.e*b.e45+a.e1*b.e145-a.e2*b.e425+a.e3*b.e345+a.e4*b.e5-a.e5*b.e4-a.e12*b.e1245+a.e13*b.e3145-a.e14*b.e15
+a.e15*b.e14-a.e23*b.e2345-a.e24*b.e25+a.e25*b.e24-a.e34*b.e35+a.e35*b.e34+a.e45*b.e-a.e123*b.e12345+a.e214*b.e125-a.e125*b.e214+a.e134*b.e315-a.e315*b.e134+a.e145*b.e1+a.e324*b.e235-a.e235*b.e324-a.e425*b.e2+a.e345*b.e3-a.e1234*b.e1253+a.e1253*b.e1234-a.e1245*b.e12+a.e3145*b.e13-a.e2345*b.e23-a.e12345*b.e123
e123 = +a.e*b.e123+a.e1*b.e23-a.e2*b.e13+a.e3*b.e12-a.e4*b.e1234+a.e5*b.e1253+a.e12*b.e3-a.e13*b.e2-a.e14*b.e324
+a.e15*b.e235+a.e23*b.e1-a.e24*b.e134+a.e25*b.e315-a.e34*b.e214+a.e35*b.e125-a.e45*b.e12345+a.e123*b.e+a.e214*b.e34-a.e125*b.e35+a.e134*b.e24-a.e315*b.e25-a.e145*b.e2345+a.e324*b.e14-a.e235*b.e15+a.e425*b.e3145-a.e345*b.e1245+a.e1234*b.e4-a.e1253*b.e5-a.e1245*b.e345+a.e3145*b.e425-a.e2345*b.e145-a.e12345*b.e45
e214 = +a.e*b.e214-a.e1*b.e24+a.e2*b.e14-a.e3*b.e1234-a.e4*b.e12+a.e5*b.e1245-a.e12*b.e4-a.e13*b.e324+a.e14*b.e2
+a.e15*b.e425-a.e23*b.e134-a.e24*b.e1+a.e25*b.e145+a.e34*b.e123-a.e35*b.e12345-a.e45*b.e125-a.e123*b.e34+a.e214*b.e+a.e125*b.e45+a.e134*b.e23+a.e315*b.e2345-a.e145*b.e25+a.e324*b.e13-a.e235*b.e3145-a.e425*b.e15+a.e345*b.e1253+a.e1234*b.e3+a.e1253*b.e345-a.e1245*b.e5-a.e3145*b.e235+a.e2345*b.e315-a.e12345*b.e35
e125 = +a.e*b.e125+a.e1*b.e25-a.e2*b.e15-a.e3*b.e1253+a.e4*b.e1245+a.e5*b.e12+a.e12*b.e5-a.e13*b.e235+a.e14*b.e425
-a.e15*b.e2-a.e23*b.e315+a.e24*b.e145+a.e25*b.e1-a.e34*b.e12345-a.e35*b.e123+a.e45*b.e214+a.e123*b.e35-a.e214*b.e45+a.e125*b.e-a.e134*b.e2345+a.e315*b.e23-a.e145*b.e24+a.e324*b.e3145+a.e235*b.e13-a.e425*b.e14-a.e345*b.e1234-a.e1234*b.e345+a.e1253*b.e3-a.e1245*b.e4+a.e3145*b.e324-a.e2345*b.e134-a.e12345*b.e34
e134 = +a.e*b.e134+a.e1*b.e34-a.e2*b.e1234-a.e3*b.e14+a.e4*b.e13+a.e5*b.e3145-a.e12*b.e324+a.e13*b.e4-a.e14*b.e3
+a.e15*b.e345+a.e23*b.e214+a.e24*b.e123-a.e25*b.e12345+a.e34*b.e1-a.e35*b.e145-a.e45*b.e315-a.e123*b.e24-a.e214*b.e23-a.e125*b.e2345+a.e134*b.e+a.e315*b.e45+a.e145*b.e35+a.e324*b.e12+a.e235*b.e1245-a.e425*b.e1253-a.e345*b.e15+a.e1234*b.e2-a.e1253*b.e425+a.e1245*b.e235-a.e3145*b.e5-a.e2345*b.e125-a.e12345*b.e25
e315 = +a.e*b.e315-a.e1*b.e35-a.e2*b.e1253+a.e3*b.e15+a.e4*b.e3145-a.e5*b.e13-a.e12*b.e235-a.e13*b.e5+a.e14*b.e345
+a.e15*b.e3+a.e23*b.e125-a.e24*b.e12345-a.e25*b.e123-a.e34*b.e145-a.e35*b.e1+a.e45*b.e134+a.e123*b.e25+a.e214*b.e2345-a.e125*b.e23-a.e134*b.e45+a.e315*b.e+a.e145*b.e34-a.e324*b.e1245+a.e235*b.e12+a.e425*b.e1234-a.e345*b.e14+a.e1234*b.e425+a.e1253*b.e2-a.e1245*b.e324-a.e3145*b.e4+a.e2345*b.e214-a.e12345*b.e24
e145 = +a.e*b.e145+a.e1*b.e45-a.e2*b.e1245+a.e3*b.e3145-a.e4*b.e15+a.e5*b.e14-a.e12*b.e425+a.e13*b.e345+a.e14*b.e5
-a.e15*b.e4-a.e23*b.e12345-a.e24*b.e125-a.e25*b.e214+a.e34*b.e315+a.e35*b.e134+a.e45*b.e1-a.e123*b.e2345+a.e214*b.e25+a.e125*b.e24-a.e134*b.e35-a.e315*b.e34+a.e145*b.e+a.e324*b.e1253-a.e235*b.e1234+a.e425*b.e12-a.e345*b.e13-a.e1234*b.e235+a.e1253*b.e324+a.e1245*b.e2-a.e3145*b.e3-a.e2345*b.e123-a.e12345*b.e23
e324 = +a.e*b.e324-a.e1*b.e1234-a.e2*b.e34+a.e3*b.e24-a.e4*b.e23+a.e5*b.e2345+a.e12*b.e134+a.e13*b.e214+a.e14*b.e123
-a.e15*b.e12345-a.e23*b.e4+a.e24*b.e3-a.e25*b.e345-a.e34*b.e2-a.e35*b.e425-a.e45*b.e235-a.e123*b.e14-a.e214*b.e13+a.e125*b.e3145-a.e134*b.e12-a.e315*b.e1245+a.e145*b.e1253+a.e324*b.e+a.e235*b.e45+a.e425*b.e35+a.e345*b.e25+a.e1234*b.e1+a.e1253*b.e145-a.e1245*b.e315+a.e3145*b.e125-a.e2345*b.e5-a.e12345*b.e15
e235 = +a.e*b.e235-a.e1*b.e1253+a.e2*b.e35-a.e3*b.e25+a.e4*b.e2345+a.e5*b.e23+a.e12*b.e315+a.e13*b.e125-a.e14*b.e12345
-a.e15*b.e123+a.e23*b.e5-a.e24*b.e345-a.e25*b.e3-a.e34*b.e425+a.e35*b.e2+a.e45*b.e324+a.e123*b.e15-a.e214*b.e3145-a.e125*b.e13+a.e134*b.e1245-a.e315*b.e12-a.e145*b.e1234-a.e324*b.e45+a.e235*b.e+a.e425*b.e34+a.e345*b.e24-a.e1234*b.e145+a.e1253*b.e1+a.e1245*b.e134-a.e3145*b.e214-a.e2345*b.e4-a.e12345*b.e14
e425 = +a.e*b.e425-a.e1*b.e1245-a.e2*b.e45+a.e3*b.e2345+a.e4*b.e25-a.e5*b.e24+a.e12*b.e145-a.e13*b.e12345-a.e14*b.e125
-a.e15*b.e214-a.e23*b.e345-a.e24*b.e5+a.e25*b.e4+a.e34*b.e235+a.e35*b.e324-a.e45*b.e2+a.e123*b.e3145+a.e214*b.e15+a.e125*b.e14-a.e134*b.e1253+a.e315*b.e1234-a.e145*b.e12-a.e324*b.e35-a.e235*b.e34+a.e425*b.e+a.e345*b.e23+a.e1234*b.e315-a.e1253*b.e134+a.e1245*b.e1+a.e3145*b.e123-a.e2345*b.e3-a.e12345*b.e13
e345 = +a.e*b.e345-a.e1*b.e3145+a.e2*b.e2345+a.e3*b.e45-a.e4*b.e35+a.e5*b.e34-a.e12*b.e12345-a.e13*b.e145-a.e14*b.e315
-a.e15*b.e134+a.e23*b.e425+a.e24*b.e235+a.e25*b.e324+a.e34*b.e5-a.e35*b.e4+a.e45*b.e3-a.e123*b.e1245+a.e214*b.e1253-a.e125*b.e1234+a.e134*b.e15+a.e315*b.e14+a.e145*b.e13-a.e324*b.e25-a.e235*b.e24-a.e425*b.e23+a.e345*b.e-a.e1234*b.e125+a.e1253*b.e214-a.e1245*b.e123+a.e3145*b.e1-a.e2345*b.e2-a.e12345*b.e12
e1234 = +a.e*b.e1234-a.e1*b.e324-a.e2*b.e134-a.e3*b.e214-a.e4*b.e123+a.e5*b.e12345+a.e12*b.e34-a.e13*b.e24+a.e14*b.e23
-a.e15*b.e2345+a.e23*b.e14-a.e24*b.e13-a.e25*b.e3145+a.e34*b.e12-a.e35*b.e1245-a.e45*b.e1253+a.e123*b.e4+a.e214*b.e3+a.e125*b.e345+a.e134*b.e2-a.e315*b.e425+a.e145*b.e235+a.e324*b.e1+a.e235*b.e145-a.e425*b.e315+a.e345*b.e125+a.e1234*b.e+a.e1253*b.e45+a.e1245*b.e35+a.e3145*b.e25+a.e2345*b.e15+a.e12345*b.e5
e1253 = +a.e*b.e1253-a.e1*b.e235-a.e2*b.e315-a.e3*b.e125+a.e4*b.e12345+a.e5*b.e123-a.e12*b.e35+a.e13*b.e25-a.e14*b.e2345
-a.e15*b.e23-a.e23*b.e15-a.e24*b.e3145+a.e25*b.e13-a.e34*b.e1245-a.e35*b.e12+a.e45*b.e1234-a.e123*b.e5-a.e214*b.e345+a.e125*b.e3+a.e134*b.e425+a.e315*b.e2-a.e145*b.e324-a.e324*b.e145+a.e235*b.e1+a.e425*b.e134-a.e345*b.e214-a.e1234*b.e45+a.e1253*b.e+a.e1245*b.e34+a.e3145*b.e24+a.e2345*b.e14+a.e12345*b.e4
e1245 = +a.e*b.e1245-a.e1*b.e425-a.e2*b.e145+a.e3*b.e12345+a.e4*b.e125+a.e5*b.e214+a.e12*b.e45-a.e13*b.e2345-a.e14*b.e25
+a.e15*b.e24-a.e23*b.e3145+a.e24*b.e15-a.e25*b.e14+a.e34*b.e1253+a.e35*b.e1234+a.e45*b.e12+a.e123*b.e345-a.e214*b.e5-a.e125*b.e4-a.e134*b.e235+a.e315*b.e324+a.e145*b.e2+a.e324*b.e315-a.e235*b.e134+a.e425*b.e1+a.e345*b.e123-a.e1234*b.e35-a.e1253*b.e34+a.e1245*b.e+a.e3145*b.e23+a.e2345*b.e13+a.e12345*b.e3
e3145 = +a.e*b.e3145-a.e1*b.e345+a.e2*b.e12345+a.e3*b.e145+a.e4*b.e315+a.e5*b.e134-a.e12*b.e2345-a.e13*b.e45+a.e14*b.e35
-a.e15*b.e34+a.e23*b.e1245+a.e24*b.e1253+a.e25*b.e1234-a.e34*b.e15+a.e35*b.e14-a.e45*b.e13-a.e123*b.e425+a.e214*b.e235-a.e125*b.e324-a.e134*b.e5-a.e315*b.e4-a.e145*b.e3-a.e324*b.e125+a.e235*b.e214-a.e425*b.e123+a.e345*b.e1-a.e1234*b.e25-a.e1253*b.e24-a.e1245*b.e23+a.e3145*b.e+a.e2345*b.e12+a.e12345*b.e2
e2345 = +a.e*b.e2345+a.e1*b.e12345+a.e2*b.e345+a.e3*b.e425+a.e4*b.e235+a.e5*b.e324+a.e12*b.e3145+a.e13*b.e1245+a.e14*b.e1253
+a.e15*b.e1234+a.e23*b.e45-a.e24*b.e35+a.e25*b.e34+a.e34*b.e25-a.e35*b.e24+a.e45*b.e23+a.e123*b.e145-a.e214*b.e315+a.e125*b.e134+a.e134*b.e125-a.e315*b.e214+a.e145*b.e123-a.e324*b.e5-a.e235*b.e4-a.e425*b.e3-a.e345*b.e2-a.e1234*b.e15-a.e1253*b.e14-a.e1245*b.e13-a.e3145*b.e12+a.e2345*b.e+a.e12345*b.e1
e12345 = +a.e*b.e12345+a.e1*b.e2345+a.e2*b.e3145+a.e3*b.e1245+a.e4*b.e1253+a.e5*b.e1234+a.e12*b.e345+a.e13*b.e425+a.e14*b.e235
+a.e15*b.e324+a.e23*b.e145+a.e24*b.e315+a.e25*b.e134+a.e34*b.e125+a.e35*b.e214+a.e45*b.e123+a.e123*b.e45+a.e214*b.e35+a.e125*b.e34+a.e134*b.e25+a.e315*b.e24+a.e145*b.e23+a.e324*b.e15+a.e235*b.e14+a.e425*b.e13+a.e345*b.e12+a.e1234*b.e5+a.e1253*b.e4+a.e1245*b.e3+a.e3145*b.e2+a.e2345*b.e1+a.e12345*b.e
Inner and Outer products
In addition to the geometric product there are two more types of multiplication used in Geometric Algebra. These extend and generalise the 'dot' and 'cross' products used in 3D vector algebra.
Inner product by a vector reduces the grade of a multivector. It is related to the dot product.
Outer product by a vector increases the grade of a multivector. It is related to the cross product.
Division
We don't tend to use the 
notation for division, since multivector multiplication is not commutative we
need to be able to distinguish between [a][b]-1 and [b]-1[a].
So instead of a divide operation we tend to multiply by the inverse.
So the problem is, how to calculate the inverse of a multivector, this
is discussed here.
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