## Linear geometry in Euclidean 4-space |

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### Contents

2Planes | 1 |

1h Anglelines of a 2plane relative to another | 22 |

Isometries in Rn and | 65 |

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2-plane containing 2-planes A(u 9 cos 9 9 sin 9 algebra angle-vector bijection Cartan's theorem Clifford parallel lines Clifford translation collinear points complex numbers consequently cross-ratio defined denote distance function elements elliptic n-space equivalent Euclidean Euclidean n-space following theorem geometry gonal homogeneous coordinates identity transformation induced inner product intersect invariant 2-planes isometry in Rn left Clifford parallel left resp left translation Lemma linear transformation linearly independent linearly independent vectors maximal set mutually isoclinic 2-planes nilpotent non-singular nonzero vector orthonormal basis pair of 2-planes plane points in Pn positive definite projective line Proof Proposition prove the following prove THEOREM quadric real numbers rectangular coordinate system reflection right translation satisfies the condition scalar semi-orthogonal sends set of mutually set of points subspace Suppose symmetric matrix trans unit pure quaternions unit quaternion unit vector vector space zero