## A treatise on the analytical geometry of the point, line, by Casey J. By Casey J.

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4 Directions and Projections In geometric algebra the notion of “direction” is given a precise mathematical representation by a “unit vector,” so the unit vectors themselves are referred to as directions. Consider the geometric product of two vectors a and b: ab = a · b + a ∧ b. 1 a || is collinear with b, and a ⊥ is perpendicular to b. The interpretations associated with the inner and outer products a ·b and a ∧b imply that 1. Vectors a and b are collinear if and only if ab = ba. 2. They are orthogonal if and only if ab = −ba.

A r ) = 0. 24) If the vectors are orthogonal, they are factors of an r -graded multivector Ar : Ar = a 1 a 2 . . a r = a 1 ∧ a 2 ∧ . . ∧ a r . 25) Then it follows that for any multivector of grade r | Ar |2 ≥ 0, if Ar = 0. 26) ˜ In the expansion of the scalar part of the product (AA), the cross terms of products of multivectors of different grades should be omitted as they have no scalar parts. Thus we have ˜ 0 = |A0 |2 + |A1 |2 + · · · + | Ar |2 ≥ 0. 23) is proved. 4 Directions and Projections In geometric algebra the notion of “direction” is given a precise mathematical representation by a “unit vector,” so the unit vectors themselves are referred to as directions.

The relative direction of M and some vector a is completely characterized by the geometric product aM = a · M + a ∧ M. 42a) a ⊥ = a ∧ MM−1 . 43a) a ⊥ M = a ∧ M = (−1) k Ma ⊥ . 39a, b). 42b) is called the rejection of vector a from the M-space. 5 C7729 C7729˙C002 Geometric Algebra and Applications to Physics Angles and Exponential Functions (as Operators) An angle is a relation between two directions. Now, following Hestenes , we give a precise mathematical expression for this relation. 44b) where i is the unit pseudoscalar of the a ∧ b-plane.