Consider the lines,

$${L_1}:{{x + 1} \over 3} = {{y + 2} \over 1} = {{z + 1} \over 2}$$

$${L_2}:{{x - 2} \over 1} = {{y - 2} \over 2} = {{z - 3} \over 3}$$

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors $$\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c $$ such that $$\widehat a\,.\,\widehat b = \widehat b\,.\,\widehat c = \widehat c\,.\,\widehat a = {1 \over 2}.$$ Then, the volume of the parallelopiped is :

The minimum of distinct real values of $$\lambda ,$$ for which the vectors $$ - {\lambda ^2}\widehat i + \widehat j + \widehat k,$$ $$\widehat i - {\lambda ^2}\widehat j + \widehat k$$ and $$\widehat i + \widehat j - {\lambda ^2}\widehat k$$ are coplanar, is

Let $$\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c $$ be unit vectors such that $${\overrightarrow a + \overrightarrow b + \overrightarrow c = \overrightarrow 0 .}$$ Which one of the following is correct ?