2.1 A New Vector Product | Geometric Algebra for Physicists

Transcript

welcome to section 2.1 called a new vector product specifically this is about geometric product but in chapter one we studied the scalar product and the outer product in two dimensions we also showed how to interpret the complex product of z and w's conjugate the scalar term in the inner product of the two vectors represents the points in the complex plane and the imaginary term records their directed area which was proven in the last chapter now the idea for a new vector product was to replace the imaginary term with the outer product and the result is the geometric product which states that for vectors a and vectors b and vector b it's equal to the scalar of the two vectors plus the outer product of the two vectors which represents that this is the sum between a scalar and a bivector now that there's no longer an imaginary term how should we view the right side of the equation how do we interpret this this actually reveals that the imaginary component is in fact a bivector we'll find that geometric algebra will eventually provide all the functions of complex arithmetic and as will also be discovered geometric algebra also encompasses quaternion now from the symmetry and anti-symmetry of the geometric products terms we can prove right here that if you reverse the order of the geometric product if you commute it then you get this negative sign in front of the outer product because this is an anti-symmetric term it thus follows that you can represent each term as a combination of the geometric product now for representing the geometric product of parallel vectors this actually ends up turning out to be a scalar which just real quick as a visualization if you remember the inner product looks at two vectors and that's not very straight but it overlays one vector onto another and then it multiplies what the magnitude of these two vectors would be giving you a scalar and so if you have vectors that are just parallel to each other then it's just the same thing as taking their magnitudes times each other which is kind of what you have visual or written down right here so the geometric product of a parallel vector only returns a scalar no bivector now the geometric product of a perpendicular vector is opposite if they're perpendicular then they're going to have no contributions for the scalar product and so they're only going to give bi vectors and so right here can represent it this orient they create an oriented plane if you multiply a if you have a first and then b right here this doesn't look perpendicular but it creates an oriented plane in this specific configuration and then right here if you switch the order then it's like this b a and then it has an opposite orientation which ends up being the negative sign and i don't know why this is here that's not actually supposed to be there but anyway this means that the orthogonal vectors are anti-commuting final thoughts are basically the geometric product is an excellent way to encode both the scalar and the outer products and is a fantastic replacement of the cross product because it can be visualized in multiple dimensions not just the third dimension so yeah it's very useful thank you for watching in the next chapter or the next section we'll be going over a basic outline of geometric algebra and what what exactly it is this was not an introduction of geometric algebra in this video it was only an introduction of the geometric product