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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.

2 Projection and rejection of vector by a bivector B. Next, we generalize the above case for a multivector M of an arbitrary grade k, which determines the ak-dimensional vector space called M-space. 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.

A vector x is said to be positively directed or negatively directed relative to the vector a according as x · a > 0 or < 0. This distinction that defines the positive and negative vectors is called the orientation or sense of the a-line. The unit vector aˆ = a |a |−1 is called the direction of the a-line, whereas aˆ gives the opposite orientation for the line. 2) where β is an arbitrary scalar. 1) by vector a gives x ∧ a = 0. 3) This is a nonparametric equation for the a-line. 3 as x ∧ aˆ = 0. 4) Now we can prove the following theorem.

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