A particle starting from the origin (0, 0) moves in a straight line in the (x, y) planes. Its coordinates at a later time are $$\left( {\sqrt 3 ,3} \right)$$. The path of the particle makes with the x-axes an angle of

The vectors $$\overrightarrow A $$ and $$\overrightarrow B $$ are such that $$\left| {\overrightarrow A + \overrightarrow B } \right| = \left| {\overrightarrow A - \overrightarrow B } \right|.$$ The angle between the two vectors is

For angles of projection of a projectile at angle (45^o $$-$$ $$\theta $$) and (45^o + $$\theta $$), the horizontal range described by the projectile are in the ratio of

If a vector $$2\widehat i + 3\widehat j + 8\widehat k$$ is perpendicular to the vector $$4\widehat j - 4\widehat i + \alpha \widehat k,$$ then the value of $$\alpha $$ is