a·(b×c) is a scalar representing the "signed volume" of a
parallelepiped (a 6-faced polyhedron all of whose faces are parallelograms lying in pairs of parallel planes)
whose dimensions and angles are given by the three vectors a, b, and c.
If the vectors represent three edges of the solid that meet at a point, and
they are named in counterclockwise order from the point of view of the
interior of the solid (i j k order, if you will), then the volume will be
positive. Otherwise negative.
The triple product is, up to sign, commutative. That
is, any ordering of a, b, and c give a result with the same absolute value,
with its sign determined by the order the edges are named. In detail,
a·(b×c) = b·(c×a) = c·(a×b),
and since a×b = -b×a, it follows that
-a·(b×c) = a·(c×b) = b·(a×c) = c·(b×a)
where det is the determinant of the square matrix formed by placing the three
vectors, one above the other.
