Velocity Vectors of a Boat Relative to the Shore and River

[s3a]

Homework Statement

The water in the river moves southwest (45 degrees south of west) at 4 mi/h.
If a motorboat is traveling due east at 15 mi/h relative to the shore, determine the speed of the boat and its heading relative to the moving water.

Homework Equations

Vector summation

The Attempt at a Solution

My attempt:
The water making the boat move 4 mi/h, 45 degrees south of west means the the water s contribution to the boat s total velocity is v_1 = <-4 cos(45deg), -4sin(45 deg)>.
Furthermore, the boat's motor's contribution to the boat's velocity is v_2 = <15,0>.
Therefore the total velocity, by vector addition, is v = v_1 + v_2 = <-4 cos(45deg), -4sin(45 deg)> + <15,0> = <-4 cos(45deg)+15, -4sin(45 deg)+0> = <-4 cos(45deg)+15, -4sin(45 deg)>.
Then, speed is the magnitude of the velocity vector. So, ||v|| = sqrt([-4 cos(45deg)+15]^2 + [-4sin(45 deg)]^2), and its direction is arctan( [-4 cos(45deg)+15] / [-4sin(45 deg)]) south of east.

Questions about my attempt:
Is the velocity I find relative to the water or shore?

I suspect it's relative to the shore and that the velocity relative to the "moving water" is simply 15 mph i^ (i hat). Am I right? If so, could someone please let me know? If not, could someone please elaborate?

Any input would be greatly appreciated!

 

[kuruman]

I recommend that you use the double subscript method for relative velocities to keep things sorted. See for example
http://physics.bu.edu/~duffy/py105/RelativeV.html