I assume you're talking about vector cross-products.
If u and v are parallel or anti-parallel, their cross-product is 0, and w is also parallel or anti-parallel to both of them, so the other two cross-products are also zero.
If u and v are not parallel, put the "tail" of u at the origin, and put the "tail" of v at the "head" of u. If you now put the "tail" of w at the "head" of v, then the "head" of w will bring you back to the origin. The three line segments defined by these vectors form a triangle that lies in a plane. The cross-products uXv, vXw, and wXu are all perpendicular to that plane, and the "right hand rule" indicates that all three have the same sense ("up" or "down"), whereas uXw would have the sense of -uXv.
But how do we know that wXu and vXw have the same magnitude? Well, we know that v = -u - w, so vXw = (-u - w)Xw = -uXw + wXw = wXu + 0. A similar argument can be formed for each of the other pairs.