Can this simple trig problem be solved using vector calculus?

[cmcraes]

I recently have been teaching myself vector calculus online, i am by no means a master but i get the general concepts. I know you can use it to solve the motion of a particle in a fluid and was curious as to whether it can be used to solve simple physics problems, involving current and wind.

Here is a sample question i got for homework:

A river flows due east at 2.32 m/s. A boat
crosses the river from the south shore to the
north shore by maintaining a constant velocity of 13.5 m/s due north relative to the water.
If the river is 322 m wide, how far downstream is the boat when it reaches the north
shore? How many degrees off course is the boat
forced by the current?
Answer in units of ◦

Using Trig my (correct) answer is 13.69789765 m and 9.7511 degrees.

Is there a way I could have found one of these answers using vector calculus? I don't care how unnecessarily complex it is, its fun!

P.S. i am familiar with line integrals so that won't be a problem
Thanks!

 

[Jolb]

Well yeah, you can do this using vector algebra, but vector algebra sort of has trigonometry built into it.

Anyway, you can treat the two velocities as vectors, say v_b is the northward velocity of the boat and v_r is the eastward velocity of the river. These vectors follow vector addition, so when the boat is moving north over the river while being dragged by the river, its overall velocity v is a vector sum:
v=v_b+v_r

If you wanted to find the angle θ that v makes with the boat's northward velocity v_b (i.e. its original northward course), you can use the formula:
v⋅v_b=|v||v_b|cosθ