Question

show that a boat move at an angle of 90degrees with respect to river in order to cross the river in a minimum time?

Answer 1

River flows with a velocity u along its path.  Let the boat have a uniform velocity v with respect to still water.

Boat wants to cross from point A on one side of the river to any point B on the other side of the river.  We want the boat to take minimum time duration.

Let the boat travel at an angle of Ф wrt to u ie. flow of river.   At any time the velocity of the boat  has two components:
   along the river flow = u + v cos Ф
   perpendicular to river flow = v sin Ф
The velocities are uniform and depend only on the angle Ф.

If the width of the river (assume to be uniform) = d, then, the time duration to travel to the other side = t = d / v sinФ
   v sin Ф is the component of velocity along the path d.
  
Hence, the time duration is minimum if the angle Ф = 90°.