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

If the angle between the vectors $$\overrightarrow A $$ and $$\overrightarrow B $$ is $$\theta $$, the value of the product $$\left( {\overrightarrow B \times \overrightarrow A } \right).\overrightarrow A $$ is equal to

Two boys are standing at the ends A and B of a ground where AB = a. The boy at B starts running in a direction perpendicular to AB with velocity v₁. The boy at A starts running simultaneously with velocity v and catches the other in a time t, where t is