Let $$OA=a,$$ $$OB=10a+2b$$ and $$OC=b$$ where $$O,A$$ and $$C$$ are non-collinear points. Let $$p$$ denote the area of the quadrilateral $$OABC,$$ and let $$q$$ denote the area of the parallelogram with $$OA$$ and $$OC$$ as adjacent sides. If $$p=kq,$$ then $$k=$$.........

A nonzero vector $$\overrightarrow a $$ is parallel to the line of intersection of the plane determined by the vectors $$\widehat i,\widehat i + \widehat j$$ and the plane determined by the vectors $$\widehat i - \widehat j,\widehat i + \widehat k.$$ The angle between $$\overrightarrow a $$ and the vector $$\widehat i - 2\widehat j + 2\widehat k$$ is ................

If $$\overrightarrow b \,$$ and $$\overrightarrow c \,$$ are two non-collinear unit vectors and $$\overrightarrow a \,$$ is any vector, then $$\left( {\overrightarrow a .\overrightarrow b } \right)\overrightarrow b + \left( {\overrightarrow a .\overrightarrow c } \right)\overrightarrow c + {{\overrightarrow a .\left( {\overrightarrow b \times \overrightarrow c } \right)} \over {\left| {\overrightarrow b \times \overrightarrow c } \right|}}\left( {\overrightarrow b \times \overrightarrow c } \right) = $$ ..............

A unit vector coplanar with $$\overrightarrow i + \overrightarrow j + 2\overrightarrow k $$ and $$\overrightarrow i + 2\overrightarrow j + \overrightarrow k $$ and perpendicular to $$\overrightarrow i + \overrightarrow j + \overrightarrow k $$ is ...........