her speed along $\vec{r}$ is 3 km/s. Not her horizontal velocity. You have to solve for that once you figure out the angle she needs to swim at to counter the current's velocity vector.
For problem #15, I do not under stand how to interpret a diagram from the information given. To my understanding this is how the diagram should look like
however it is the hypotenuse of the triangle that represents the women's speed in still water, why????
Also, it says she wants to go directly opposite from her starting point, so the horizontal part of the triangle should represent her speed, am I right?
Thank you for your time.
pick a coordinate system. Usually in navigation problems north is positive y, east is positive x. Orient the river on the y axis.
Now write down the vectors for the river current, her swimming velocity, and the x and y components of her swimming velocity in terms of theta.
Now adjust theta so that the sum of her y component of velocity and the current velocity is zero.
Use that theta to compute horizontal velocity.