\(\vec B\times \vec C\) = BCsin θ\(\hat n\)[\(\hat n\) = unit vector perpendicular to the plane contining \(\vec B\) and \(\vec C\)]

\(\vec A\,\times(\vec B\times\vec C)=\vec A\times\hat n(BC\,sin\,\theta)=(BC\, sin\,\theta)A\,sin\alpha\,\hat P\)

[\(\hat P\) = unit vector perpendiculat to the plane containing \(\vec A\,and\,(\vec B\times\vec C)\) ]

Hence, \(\vec A\times(\vec B\times \vec C)\) will lie in the plane of \(\vec B\) and \(\vec C\), and is perpendicular to \(\vec A\).