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a e F a₁ a₁i a₂ addition and multiplication angle aşk b₁ b₂ b₁i b₂j basis of R3 called cancellation law Cauchy's inequality commutative law complex numbers components coordinate plane coordinates a1 cross product defined DEFINITION denoted direction dot product Euclidean space Example EXERCISES field of scalars Figure finite-dimensional given Hence homomorphism inner product inner product space isomorphism let x1 linearly independent mapping n-tuples nonzero vector normal orthogonal basis normal vector ordered field parallelogram parametric equations perpendicular Proof R₂ ra₂ real numbers respective coordinates S₁ S₂ scalar multiplication shown in Fig Similarly Solution span R3 terminal point tetrahedron THEOREM triangle u₁ u₂ unique unit vector unitary space V₂ V2 or V3 vector algebra vector space vectors X1 x-axis x₁ y₁ Σ Σ