If the vectors $$\vec c,\,\vec a = x\hat i + y\hat j + z\hat k$$     and $$\hat b = \hat j$$  are such that $$\vec a,\,\vec c$$  and $${\vec b}$$ form aright handed system then $${\vec c}$$ is :

Solution :
Since $$\vec a,\,\vec c,\,\vec b$$  form a right handed system,
\[\therefore \,\,\,\vec c = \,\vec b \times \vec a = \left| \begin{array}{l} \hat i\,\,\,\,\,\hat j\,\,\,\,\,\hat k\\ 0\,\,\,\,\,1\,\,\,\,\,0\\ x\,\,\,\,\,y\,\,\,\,z \end{array} \right| = z\hat i - x\hat k\]