We want to do linear transformations such as rotations, translations, etc. We can do this by using Geometric Algebra or by other mathematical tools such as matrices, the advantage of Geometric Algebra is that such operations can be handled in a general way without the need to specify an arbitrary coordinate system and without all the index notation involved with matrices. However no one type of algebra is perfect and so it might be good if we can work out a way to transfer back and forward between the these representations.
Imagine that there is an absolute coordinate system and the basis vectors are defined in terms of these absolute coordinates so, taking a 3D case we get,
e1 = (e1x,e1y,e1z)
- e1x = x coordinate, in absolute coordinates, of e1.
- e1y = y coordinate, in absolute coordinates, of e1.
- e1z = z coordinate, in absolute coordinates, of e1.
and similarly for the other basis vectors:
e2 = (e2x,e2y,e2z)
e3 = (e3x,e3y,e3z)
So what is e1^e2 or e1?e2 ?
Starting with the 2D case there is only one bivector, this represents the volume enclosed by the two vectors:
e1^e2 = e1x e2y - e2x e1y = det[matrix formed from basis vectors]
This can be extended to the general n-dimensional case where the pseudoscalar is the determinant of the matrix formed from all the basis vectors.
So what about the bivectors of 3D vectors? I would like to try to relate this to the minor of the determinant, that is the determinant left when an element (together with its row and column) is removed.
e1^e2 = (e1x,e1y,e1z)^(e2x,e2y,e2z)
In the geometric interpretation we can see that in 3D the bivector represents the volume enclosed by the two vectors in the plane of the vectors, since we are in 3D, this is equivalent to the cross product so,
(e1^e2)x = e1y * e2z - e2y * e1z = minor of e3x
(e1^e2)y = e1z * e2x - e2z * e1x = minor of e3y
(e1^e2)z = e1x * e2y - e2x * e1y = minor of e3z
So, perhaps we calculate the bivector by drawing the matrix formed by putting the basis vectors side-by-side, then taking its minor by removing the row associated with its own coordinate type and removing the column of the basis vector not associated with.
We have to be very careful with signs as the sign alternates with terms as follows:
To choose bivectors for 3D multivectors we start with the psudoscalar e1^e2^e3 which represents the whole determinant. We then split it up, taking into account the sign, as above:
e1^e2^e3 = e1 * e2^e3 - e2 * e1^e3 + e3 * e1^e2
Where the sign is negative then we invert the order, which gives the basis bivectors as follows:
- e2^e3 is the minor of e1
- e3^e1 is the minor of e2
- e1^e2 is the minor of e3