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Definition: If u = [u_1, u_2, … , u_n] and v = [v_1, v_2, … , v_n] , then the dot/scalar product u * v of u and v is defined by u * v = (u_1)(v_1) + (u_2)(v_2) + ... + (u_n)(v_n).
Source: Linear Algebra: A Modern Introduction, 3rd edition by David Poole (note-custom edition titled Matrix Algebra)