If $$X.A=0, X.B=0, X.C=0$$ for some non-zero vector $$X,$$ then $$\left[ {A\,B\,C} \right] = 0$$

A vector $$\overrightarrow A $$ has components $${A_1},{A_2},{A_3}$$ in a right -handed rectangular Cartesian coordinate system $$oxyz.$$ The coordinate system is rotated about the $$x$$-axis through an angle $${\pi \over 2}.$$ Find the components of $$A$$ in the new coordinate system in terms of $${A_1},{A_2},{A_3}.$$

The area of the triangle whose vertices are $$A(1, -1, 2), B(2, 1, -1), C(3, -1, 2)$$ is ..........

The unit vector perpendicular to the plane determined by $$P\left( {1, - 1,2} \right),\,Q\left( {2,0, - 1} \right)$$ and $$R\left( {0,2,1} \right)$$ is ...........