The cross product of a vector with a cross product is called the triple cross product.

The expansion formula of the triple cross product or Lagrange's formula is $$\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b}-(\vec{a} \cdot \vec{b})\vec{c}$$ (``exterior dot far times near minus exterior dot near times far'' -- this works also when ``exterior'' is the last factor).

The formula shows that this vector is in the plane spanned by the vectors $\vec{b}$ and $\vec{c}$ (when these are not parallel).

Note that the use of parentheses in the triple cross products is necessary, since the cross product operation is not associative, i.e., generally we have $$(\vec{a}\times\vec{b})\times\vec{c} \neq\vec{a}\times(\vec{b}\times\vec{c})$$ (for example: $(\vec{i}\times\vec{i})\times\vec{j} = \vec{0}$ but $\vec{i}\times(\vec{i}\times\vec{j}) = -\vec{j}$ when $(\vec{i},\,\vec{j},\,\vec{k})$ is a right-handed orthonormal basis of $\mathbb{R}^3$ ). So the system $(\mathbb{R}^3,\,+,\,\times)$ is not a ring.

A direct consequence of the expansion formula is the Jacobi identity $$\vec{a}\times(\vec{b}\times\vec{c})+\vec{b}\times(\vec{c}\times\vec{a})+ \vec{c} \times(\vec{a}\times\vec{b}) = \vec{0},$$ which is one of the properties making $(\mathbb{R}^3,\,+,\,\times)$ a Lie algebra.

It follows from the expansion formula also that $$(\vec{a}\times\vec{b})\times(\vec{c}\times\vec{d}) = (\vec{a}\vec{b}\vec{d})\vec{c}-(\vec{a}\vec{b}\vec{c})\vec{d}$$ where $(\vec{u}\vec{v}\vec{w})$ means the triple scalar product of $\vec{u}$ , $\vec{v}$ and $\vec{w}$ .