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addition and multiplication basis of R3 cancellation law Cauchy's inequality commutative commutative law complex numbers components coordinate plane coordinate system cross product defined denoted diagonals direction dot product Euclidean space Example EXERCISES field of scalars Figure finite-dimensional Hence homomorphism identity element initial point inner product space intersection isomorphism length Line on point linear combination linearly independent mapping medians n-dimensional n-tuples nonzero vector normal orthogonal basis normal vector operations of addition ordered field parallelogram perpendicular plane passing product in V2 Proof proper subspace Prove vectorially real numbers respective coordinates scalar multiplication shown in Fig Similarly Solution span R3 subset terminal point tetrahedron THEOREM triangle unique unit vector unitary space V2 and R2 V2 and V3 V2 or V3 vector algebra vector space vectors of V2 vectors xi x-axis xu x2