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3-space a x b ABCD affine geometry affine space affine transformation algebraic angle arbitrary Axiom Axioms OP1 axis barycentric coordinates basis vectors boundary center of mass coefficients collinear column compute congruent curve defined definition dependent determinant dilatation dimension direction distance Euclidean transformation Example Exercises FIGURE fixed point formula geometric given Hence independent inner product inverse length Let ex linear transformation mass-points matrix method mid-point multiply mxPx notion obtain orientation origin orthogonal projection orthogonal transformation orthonormal outer product parallel parallelepiped parallelogram perpendicular plane polygon Proof Prove ratio reader real numbers reflection result rotation scalar Section 2.7 segment Similarly spanned subspace Suppose symmetry tetrahedron translation triangle ABC unique vector area vector space vectors fx velocity vertices volume