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a₁ a₁i a₂ angle associative assume b₁ b₂ basis called Chapter commutative complex numbers components consider coordinates coordinates a1 cross product defined DEFINITION denoted dependent dimension direction distance dot product easily elements equal equation Euclidean space evidently Example EXERCISES exists fact field Figure follows geometric given Hence hold identity inner product intersection isomorphism length linearly independent mapping midpoint nonzero vector normal orthogonal basis normal vector obtain operation parallelogram passing perpendicular plane positive Proof properties Prove R₂ real numbers result S₁ S₂ scalar multiplication segments shown in Fig Similarly Solution span subspace THEOREM triangle u₁ unique unit vector V₂ vector algebra vector space vectors X1 y₁ zero