Editing Geometric algebra (section)

==Geometric Algebra Intro==
Geometric Algebra is the Clifford Algebra of a vector space over the field of real numbers endowed with a quadratic form. It's an elegant mathematical framework for expressing geometrical ideas and doing computations in fields such as physics, engineering and in computer vision (see [https://www.eleceng.adelaide.edu.au/personal/dabbott/wiki/index.php/Semester_B_Final_Report_2013_-_Secure_communications_without_key_exchange%3F_2013#Geometric_Algebra Final Report GA] for a list of some research applications). It correctly expresses the properties of physical space and one of the main benefits for our project was that it could be extended effortlessly to spaces of arbitrary dimensions.

[[Image:DefiningVectors.jpg|center|750px]]
Vectors are defined as shown on the left and are known as Cartesian coordinates, defined by Descartes in 1637. In 1879, Clifford's further described the space by adopting the folllowing-

[[Image:GAExp1.png|center|910px]]


[[Image:GAExp2.png|center|1200px]]


[[Image:GAExp3.png|center|320px]]


The 3D space is then simply defined with each plane being defined by the multiple of two of e1, e2 and e3 and the trivector i represents the volume in its entirety. As a result of the anti communicating  characteristics, we have '''Dual Representation''' of each bivector, by utilising the trivector to give.... '''e2e3 = ie1''',''' e3e1 = ie2''' and '''e1e2 = ie3'''.