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6 votes

Thread map to 2 lists of inputs, including the nondiagonal terms

a = {a1, a2, a3}; b = {b1, b2, b3}; f @@@ Tuples[{a, b}] (* {f[a1, b1], f[a1, b2], f[a1, b3], f[a2, b1], f[a2, b2], f[a2, b3], f[a3, b1], f[a3, b2], f[a3, b3]} *)
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