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Scalar Wave Benefits

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Let V be the set of all ordered pairs of real numbers, and consider the following vector addition and scalar multiplication operations on u = (U1, U2) and v = (V1, V2): u v = (U1 V1, U2 V2), ku = (ku1, ku2). Is there a zero vector in V? If so, what is it? For a given vector u ∈ V, is there a vector  u ∈ V such that u ( u) = 0? If so, what is it? Find all axioms, if any, from the list of 10 axioms for real vector spaces that fail to hold and prove that they are not holding.