The correct way to do this is with a POLAR PLOT, which is like a compass
rose, only with inner rings that are evenly spaced from the center. Please see the attached file so you can see how I calculated my answers (although I'm not sure if it got uploaded properly).
1. Draw a line from the center of the plot to mark the vessel's course of 63º, and make it into an arrow that stops at the fifth ring since his speed is 5 knots. Label this line as "Vessel."
2. Draw a line from the center marking his bearing of 136º. Label this line as "Bearing."
3. Draw a line parallel to the bearing line that intersects the tip of the vessel's vector at the fifth ring. Label this line as "DRM" (DIRECTION OF RELATIVE MOVEMENT). The reason for this is because in order to collide with an object, the bearing to that object must remain constant. (Think about it.)
4. Since your speed is 13 knots, find the point on the DRM line where the 13th ring intercepts it. That's kind of a trick question because there are two points on the DRM line where the 13th ring intercepts it. Draw a line from the center to each of these two points. The line closest to the bearing line (forming an acute angle to the bearing line) is the COURSE TO INTERCEPT, which is 114º -- the answer to our question. Mark this line as "CTI." The other line that's further away from the bearing line (forming an obtuse angle to the bearing line) is the COURSE TO EVADE, which is 336º. Mark this line as "CTE."
The first part of the question doesn't ask how to evade the ship, but the second part of the question asks how to evade a hurricane
. You'd use the same principles to answer the second part of the question, only instead of calculating the Course to Intercept, you'd calculate the Course to Evade.
Also, the length of the DRM line between the vessel's vector and your CTI vector is the CLOSING SPEED. You can use your dividers to measure it relative to the distance in the rings. The TIME TO INTERCEPT is calculated by dividing the Distance to the vessel by the Closing Speed.
Please note: If you're looking at the file I uploaded (assuming it shows up here somewhere), it's just a coincidence that the polar plot I used has exactly 13 lines, the same speed as my vessel (13 knots), which puts my CTI and CTE vectors at the intersection of the outer ring. If the vessel were traveling at 14 knots, let's say, I'd make each ring equal to 2 nautical miles rather than 1 nautical mile. As a result, I would draw the vector indicating my direction and speed of 14 knots out to the seventh ring, not the 14th, since there is none. And I would draw the vessel's direction and speed of 5 knots out to half-way between the second and third rings (5 knots divided by 2 is 2.5).
I hope this helps someone. I was taught this method in a captain's class in preparation for my six-pack test, and I'm fairly certain that's how the pros do it. Once you get the hang of it, it's really very simple and only takes a matter of seconds to draw 4 quick lines on the plotter.