The problem and its solution are attached as TheProblemAndSolution.jpg.
Vector addition. Cross product.
The Attempt at a Solution
Given that I am just supposed to take Fig. 1 for what it is based on the way the problem is phrased, my biggest confusion is in understand how Fig. 1 was translated to Fig. 2 as well as in interpreting Fig. 2.
Firstly, I feel it shouldn't matter what the orientation of the boat is since the forces we are considering in the diagram will have the same magnitudes and they will all be pointing in the same directions with respect to each other. More specifically, defining the axes to be aligned with the orientation of the boat, the 20 lb force on the sail seems to me like it should be 5 feet upward in the same direction as the i_3 vector while directed in the direction of the i_2 vector but it should not be 6 feet from the tail of the i_2 vector. The other 20 lb force vector's tail seems to me like it should be placed on the top-right corner of the bottom face of the rectangular prism directed toward the opposite direction of the i_2 vector instead of it being translated 6 feet in the direction opposite the i_2 vector from where I feel it shoul dbe. As for the 10 lb force pointing downward, I agree with where it is. For the upward pointing 10 lb force vector, I also feel it should be on the top-right of the bottom face of the rectangular prism but, unlike 20 lb force vector that is not applied to the sail, it should be pointing in the direction of the i_3 vector.
Looking online, I found: (1) “In mechanics, a couple is a system of forces with a resultant (a.k.a. net, or sum) moment but no resultant force.” and (2) “Another term for a couple is a pure moment”.
What I found online makes sense to me so I get what a couple vector is and I also get the algebra in computing it. That is, it's the non-zero vector sum of all the torques (applied per force) where the net force is zero.
To me, the r_n and f_n numbers obtained for the 10 lb and 20 lb forces do not seem consistent with the information provided by Fig. 1.2.
If you need me to say more, just ask.
Any help in solving this problem would be greatly appreciated!
Thanks in advance!