The 'right hand rule'.
Let a and b be vectors in 3-dimensional Euclidian space (R3) such that a has components a1, a2, a3, b has components b1, b2, b3. Then the cross product a x b is defined as:
a x b = (a2b3 - a3b2)i - (a1b3 - a3b1)j + (a1b2 - a2b1)k
where i, j, and k are the unit basis vectors of R3. a x b can be found perpendicular to b and a by use of the dot product.
Now let R3 be re-coordinatized in terms of a, b, and a x b. This will result in the translation of those vectors into the unit basis vectors. From there, we see that:
a x b = i x j = kb x a = j x i = -k
which clearly obeys the 'right hand rule' - just hold up your hand and see. This is concurrently analogous to the uncoordinatized vectors. As any two vectors and their cross product in R3 may be processed in this manner, the cross product obeys the right hand rule and its result is perpendicular to its basis vectors.
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