Try expanding out each vector in components, A = Ax(x) + Ay(y) + Az(z), and reorganizing to get your desired result. For example,
A.(B X C) = A.[(ByCz - BzCy)(x) + (BzCx - BxCz)(y) + (BxCy - ByCx)(z)]
= AxByCz - AxBzCy + AyBzCx - AyBxCz + AzBxCy - AzByCx
= Bx(CyAz - CzAy) + By(CzAx - CxAz) + Bz(CxAy - CyAx)
= B.(C X A)
For the components of a cross product it's helpful to remember the following. Consider the cross product Q = R X S, which gives a vector Q. The x-component Q will involve cross terms of the non-x components of R and S, ie. RySz and RzSy. The positive term will be cyclic in (x,y,z) and the negative will be anticyclic. So, for example, in (RySz - RzSy)(x), the first term RySz(x) is cyclic (y,z,x), while the second term is anticyclic (z,y,x). You can verify this in the cross products above.