R^3 is the set of all triples (or 3-tuples) of real numbers: RxRxR. an element of R^3 is usually written like so: (a,b,c). it is made into a vector space by the addition:

(x1,x2,x3) + (y1,y2,y3) = (x1+y1,x2+y2,x3+y3), or coordinate-wise addition, and the scalar (scaling) multiplication:

a(x1,x2,x3) = (ax1,ax2,ax3).