Let $$\widehat u = {u_1} \widehat i + {u_2}\widehat j + {u_3}\widehat k$$ be a unit vector in $${{R^3}}$$ and
$$\widehat w = {1 \over {\sqrt 6 }}\left( {\widehat i + \widehat j + 2\widehat k} \right).$$ Given that there exists a vector $${\overrightarrow v }$$ in $${{R^3}}$$ such that $$\left| {\widehat u \times \overrightarrow v } \right| = 1$$ and $$\widehat w.\left( {\widehat u \times \overrightarrow v } \right) = 1.$$ Which of the following statement(s) is (are) correct?

Let $$\Delta PQR$$ be a triangle. Let $$\vec a = \overrightarrow {QR} ,\vec b = \overrightarrow {RP} $$ and $$\overrightarrow c = \overrightarrow {PQ} .$$ If $$\left| {\overrightarrow a } \right| = 12,\,\,\left| {\overrightarrow b } \right| = 4\sqrt 3 ,\,\,\,\overrightarrow b .\overrightarrow c = 24,$$ then which of the following is (are) true?

Let $$\overrightarrow x ,\overrightarrow y $$ and $$\overrightarrow z $$ be three vectors each of magnitude $$\sqrt 2 $$ and the angle between each pair of them is $${\pi \over 3}$$. If $$\overrightarrow a $$ is a non-zero vector perpendicular to $$\overrightarrow x $$ and $$\overrightarrow y \times \overrightarrow z $$ and $$\overrightarrow b $$ is a non-zero vector perpendicular to $$\overrightarrow y $$ and $$\overrightarrow z \times \overrightarrow x ,$$ then

The vector (s) which is/are coplanar with vectors $${\widehat i + \widehat j + 2\widehat k}$$ and $${\widehat i + 2\widehat j + \widehat k,}$$ and perpendicular to the vector $${\widehat i + \widehat j + \widehat k}$$ is/are