Create vector set

Question

I would want to create the vectors $(\pm x_1, \pm x_2, \ldots,\pm x_n), \; \{x_1,x_2,x_3 \in \mathbb R\}, \{n\in \mathbb N \}$.

For small $n$, this can be done manually (here $n=2, \, x_1 = 1, \, x_2 = 2$),

$(\pm 1, \pm 2) = \{(-1, -2), \;(1,-2), \; (-1,2), \;(1,2) \}$

but the number of vectors increases exponentially. Is there a neat built-in syntax for this kind of list manipulation?

Answer 1

Like Coolwater pointed out in the comments you could use the built-in Tuples command:

Tuples[{-#,#}&/@Range[1,n]]

which works pretty fast.

Another (slow) but natural approach is using PatternMatching:

n=2;
Flatten[Table[PlusMinus[i],{i,1,n}]//.{a___,PlusMinus[b_],c___}:>{{a,b,c},{a,-b,c}},n-1]

{{1,2},{1,-2},{-1,2},{-1,-2}}