× (multiplication sign)
× is an operator that serves two distinct functions:

It is the cross product of vectors in ℝ³ or ℂ³. In this scenario, it is a binary operator.

It is the Cartesian product of sets. In this scenario, it is an nary infix operator.
Syntax

u × v

u
andv
are threedimensional vectors


S1 × S2 × ⋯ × Sn

S1
,S2
, ...,Sn
are sets

Description and examples
Vector cross product
If u
and v
are threedimensional (real or complex) vectors, then u × v
is the vector cross product of u
and v
.
The product is complex iff at least one of the operands is complex.
a ≔ ❨4, 1, 2❩; b ≔ ❨1, 0, 2❩;
a × b
⎛2 ⎞ e⎜−6⎟ ⎝−1⎠
a ≔ ❨4, 1, 2❩; b ≔ ❨1, 0, 2❩; c ≔ ❨−2, 1, −1❩;
(a  b × c)
−9
Cartesian product
If S1
, S2
, ..., Sn
are sets, then S1 × S2 × ⋯ × Sn
is the Cartesian product of S1
, S2
, ..., Sn
.
{1, 2, 3} × {"a", "b", "c"} × {"!", "?"}
{(3, a, !), (1, c, ?), (2, a, ?), (1, b, ?), (3, c, !), (2, c, ?), (3, b, !), (1, a, !), (3, a, ?), (2, b, ?), (1, c, !), (2, a, !), (3, c, ?), (1, b, !), (3, b, ?), (1, a, ?), (2, c, !), (2, b, !)}
animals ≔ { "dog", "rat", "cat", "rabbit", "guinea pig" };
sexes = { "male", "female" }
{(dog, female), (rat, male), (guinea pig, male), (cat, female), (rat, female), (rabbit, male), (guinea pig, female), (rabbit, female), (dog, male), (cat, male)}
Notes
The ×
operator is implemented by the cross
function. CrossProduct
is a distinct function that is equivalent to the restriction of cross
to (two) vector operands.