Let $$\overrightarrow u $$, $$\overrightarrow v $$ and $$\overrightarrow w $$ be vectors in three-dimensional space, where $$\overrightarrow u $$ and $$\overrightarrow v $$ are unit vectors which are not perpendicular to each other and $$\overrightarrow u $$ . $$\overrightarrow w $$ = 1, $$\overrightarrow v $$ . $$\overrightarrow w $$ = 1, $$\overrightarrow w $$ . $$\overrightarrow w $$ = 4

If the volume of the paralleopiped, whose adjacent sides are represented by the vectors, $$\overrightarrow u $$, $$\overrightarrow v $$ and $$\overrightarrow w $$, is $$\sqrt 2 $$, then the value of $$\left| {3\overrightarrow u + 5\overrightarrow v } \right|$$ is ___________.

Let $$\overrightarrow a = 2\widehat i + \widehat j - \widehat k$$ and $$\overrightarrow b = \widehat i + 2\widehat j + \widehat k$$ be two vectors. Consider a vector c = $$\alpha $$$$\overrightarrow a$$ + $$\beta $$$$\overrightarrow b$$, $$\alpha $$, $$\beta $$ $$ \in $$ R. If the projection of $$\overrightarrow c$$ on the vector ($$\overrightarrow a$$ + $$\overrightarrow b$$) is $$3\sqrt 2 $$, then the
minimum value of ($$\overrightarrow c$$ $$-$$($$\overrightarrow a$$ $$ \times $$ $$\overrightarrow b$$)).$$\overrightarrow c$$ equals ................

Let a and b be two unit vectors such that a . b = 0. For some x, y$$ \in $$R, let $$\overrightarrow c = x\overrightarrow a + y\overrightarrow b + \overrightarrow a \times \overrightarrow b $$. If | $$\overrightarrow c $$| = 2 and the vector c is inclined at the same angle $$\alpha $$ to both a and b, then the value of $$8{\cos ^2}\alpha $$ is ..............

Suppose that $$\overrightarrow p ,\overrightarrow q $$ and $$\overrightarrow r $$ are three non-coplanar vectors in $${R^3}$$. Let the components of a vector $$\overrightarrow s $$ along $$\overrightarrow p ,$$ $$\overrightarrow q $$ and $$\overrightarrow r $$ be $$4, 3$$ and $$5,$$ respectively. If the components of this vector $$\overrightarrow s $$ along $$\left( { - \overrightarrow p + \overrightarrow q + \overrightarrow r } \right),\left( {\overrightarrow p - \overrightarrow q + \overrightarrow r } \right)$$ and $$\left( { - \overrightarrow p - \overrightarrow q + \overrightarrow r } \right)$$ are $$x, y$$ and $$z,$$ respectively, then the value of $$2x+y+z$$ is