If $$\overrightarrow a ,$$ $$\overrightarrow b $$ and $$\overrightarrow c $$ are three non coplanar vectors, then
$$\left( {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right).\left[ {\left( {\overrightarrow a + \overrightarrow b } \right) \times \left( {\overrightarrow a + \overrightarrow c } \right)} \right]$$ equals

Let $$a, b, c$$ be distinct non-negative numbers. If the vectors $$a\widehat i + a\widehat j + c\widehat k,\widehat i + \widehat k$$ and $$c\widehat i + c\widehat j + b\widehat k$$ lie in a plane, then $$c$$ is

Let $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c ,$$ be three non-coplanar vectors and $$\overrightarrow p ,\overrightarrow q ,\overrightarrow r,$$ are vectors defined by the relations $$\overrightarrow p = {{\overrightarrow b \times \overrightarrow c } \over {\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}},\,\,\overrightarrow q = {{\overrightarrow c \times \overrightarrow a } \over {\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}},\,\,\overrightarrow r = {{\overrightarrow a \times \overrightarrow b } \over {\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}}$$ then the value of the expression $$\left( {\overrightarrow a + \overrightarrow b } \right).\overrightarrow p + \left( {\overrightarrow b + \overrightarrow c } \right).\overrightarrow q + \left( {\overrightarrow c + \overrightarrow a } \right),\overrightarrow r $$ is equal to

The number of vectors of unit length perpendicular to vectors $$\overrightarrow a = \left( {1,1,0} \right)$$ and $$\overrightarrow b = \left( {0,1,1} \right)$$ is