Two billiard balls are rolling on a flat table. One has velocity components  and other has velocity components   If both the balls start moving from the same point, then what is the angle between their paths?
Let  and  be the angles that the two balls make with X-axis.
Given, the velocity components of first ball are,Â
Â
Â
Therefore, angle is given by,Â
        Â
       Â
The velocity components of second ball are,
 ,
∴ angle is given by, Â
         Â
           Â
The angle between the paths of two ball is equal to the angle between their velocity vectors.
∴  The angle between the paths of two balls is,
         Â
Let us resolve the rectangular components of a vector  represented by  in three dimensions. From P, draw a perpendicular PM on XY plane and from M draw a perpendicular PM on XY plane and from M draw perpendicular ML on Y axis and MK on X axis. Using triangle law toÂ
       Â
Here, Â Â Â along NE