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 ...........

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

If $$A, B, C, D$$ are any four points in space, prove that -
$$\left| {\overrightarrow {AB} \times \overrightarrow {CD} + \overrightarrow {BC} \times \overrightarrow {AD} + \overrightarrow {CA} \times \overrightarrow {BD} } \right| = 4$$ (area of triangle $$ABC$$)

The sides of a triangle inscribed in a given circle subtend angles $$\alpha $$, $$\beta $$ and $$\gamma $$ at the centre. The minimum value of the arithmetic mean of $$cos\left[ {\alpha + {\pi \over 2}} \right],\,\cos \left[ {\beta + {\pi \over 2}} \right]$$ and $$cos\left[ {\gamma + {\pi \over 2}} \right]$$ is equal to _______.