Let $$\mathop a\limits^ \to = 3\mathop i\limits^ \wedge + 2\mathop j\limits^ \wedge + x\mathop k\limits^ \wedge $$ and $$\mathop b\limits^ \to = \mathop i\limits^ \wedge - \mathop j\limits^ \wedge + \mathop k\limits^ \wedge $$ , for some real x. Then $$\left| {\mathop a\limits^ \to \times \mathop b\limits^ \to } \right|$$ = r is possible if :

Let $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be three unit vectors, out of which vectors $$\overrightarrow b $$ and $$\overrightarrow c $$ are non-parallel. If $$\alpha $$ and $$\beta $$ are the angles which vector $$\overrightarrow a $$ makes with vectors $$\overrightarrow b $$ and $$\overrightarrow c $$ respectively and $$\overrightarrow a $$ $$ \times $$ ($$\overrightarrow b $$ $$ \times $$ $$\overrightarrow c $$) = $${1 \over 2}\overrightarrow b $$, then $$\left| {\alpha - \beta } \right|$$ is equal to :

The sum of the distinct real values of $$\mu $$, for which the vectors, $$\mu \widehat i + \widehat j + \widehat k,$$   $$\widehat i + \mu \widehat j + \widehat k,$$   $$\widehat i + \widehat j + \mu \widehat k$$  are co-planar, is :

Let $$\sqrt 3 \widehat i + \widehat j,$$    $$\widehat i + \sqrt 3 \widehat j$$  and   $$\beta \widehat i + \left( {1 - \beta } \right)\widehat j$$ respectively be the position vectors of the points A, B and C with respect to the origin O. If the distance of C from the bisector of the acute angle between OA and OB is $${3 \over {\sqrt 2 }}$$, then the sum of all possible values of $$\beta $$ is :