Let  be the velocities of boat and river water respectively.
Let  be the resultant velocity of boat.
Boat heads towards north and water current is in the direction  east of south.
Therefore, the angle between  isÂ
Now using parllelogram law of vector addition, the magnitude of resultant velocity  is ,
 Â
   Â
    Â
Let the resultant velocity vector subtend an angle  with the north.
Then,
     Â
         Â
This implies,
          Â
Let a given vector  be resolved inÂ
Hence both the sets of components are identical. There is one and only one set of resolved components in a particular direction.
Let A and B be the magnitude of two vectors  and let A>B.
When  act in opposite direction, the magnitude of resultant is A - B
Now,
        A - B  = 7 [Given]        ...(1)
When  act perpendicular to each other, then their resultant is,
            Â
                  ...(2)
Now, Â Â Â Â Â Â Â
∴          Â
and      Â
              Â
or                    ...(3)
On solving equation (1) and (3), we get
        A = 12 and B = 5