Let $$\overrightarrow a $$ , $$\overrightarrow b $$ and $$\overrightarrow c $$ be three unit vectors such that
$$\overrightarrow a + \vec b + \overrightarrow c = \overrightarrow 0 $$. If $$\lambda = \overrightarrow a .\vec b + \vec b.\overrightarrow c + \overrightarrow c .\overrightarrow a $$ and
$$\overrightarrow d = \overrightarrow a \times \vec b + \vec b \times \overrightarrow c + \overrightarrow c \times \overrightarrow a $$, then the ordered pair, $$\left( {\lambda ,\overrightarrow d } \right)$$ is equal to :

A vector $$\overrightarrow a = \alpha \widehat i + 2\widehat j + \beta \widehat k\left( {\alpha ,\beta \in R} \right)$$ lies in the plane of the vectors, $$\overrightarrow b = \widehat i + \widehat j$$ and $$\overrightarrow c = \widehat i - \widehat j + 4\widehat k$$. If $$\overrightarrow a $$ bisects the angle between $$\overrightarrow b $$ and $$\overrightarrow c $$, then:

Let $$\alpha $$ $$ \in $$ R and the three vectors

$$\overrightarrow a = \alpha \widehat i + \widehat j + 3\widehat k$$, $$\overrightarrow b = 2\widehat i + \widehat j - \alpha \widehat k$$

and $$\overrightarrow c = \alpha \widehat i - 2\widehat j + 3\widehat k$$.

Then the set S = {$$\alpha $$ : $$\overrightarrow a $$ , $$\overrightarrow b $$ and $$\overrightarrow c $$ are coplanar} :

If the volume of parallelopiped formed by the vectors $$\widehat i + \lambda \widehat j + \widehat k$$, $$\widehat j + \lambda \widehat k$$ and $$\lambda \widehat i + \widehat k$$ is minimum, then $$\lambda $$ is equal to :