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 ...........

Given that $$\overrightarrow a = \left( {1,1,1} \right),\,\,\overrightarrow c = \left( {0,1, - 1} \right),\,\overrightarrow a .\overrightarrow b = 3$$ and $$\overrightarrow a \times \overrightarrow b = \overrightarrow c ,$$ then $$\overrightarrow b \, = $$.........

The components of a vector $$\overrightarrow a $$ along and perpendicular to a non-zero vector $$\overrightarrow b $$ are ......and .....respectively.

Let $$b = 4\widehat i + 3\widehat j$$ and $$\overrightarrow c $$ be two vectors perpendicular to each other in the $$xy$$-plane. All vectors in the same plane having projecttions $$1$$ and $$2$$ along $$\overrightarrow b $$ and $$\overrightarrow c, $$ respectively, are given by ...........