Rockin Robbins encouraged me to post in this thread, so here's my take on a vector analysis attack
as seen here on the last picture 
I made a video that shows the whole process from target spottet, till target on crash dive
->
(Don't get confused that I chose 90° for my approach, you can do this with any angle! It's not limited to 90° like seen on the Dick O'Kane Method picture. This is the method in the last picture, which of course is similar, but there are some important differences.)
In case the video leaves some questions, here's what happens, step by step:
Determine the target's speed: There are many ways to do this. The one I choose in this video is the so called "3 minute rule". You mark the current position of the target on the map, then you wait 3 minutes and mark again. The distance beween those two marks in yards represents the target speed. In the video the two marks are 900yds away from each other, that means the target runs at 9kts.
(In the video the "no map contact update" realism setting is disabled, so the marks are pretty straightforward. However, usually I have that option enabled, so there's no contact updates on my map. It's still pretty easy to mark the target on the map without those automatic updates, you should try it, it's no big deal and adds to the immersion if you like micromanagement. 
All you have to do is to get target bearing and distance and draw it on your map, maybe I'll make a tutorial Video about that too )
Planning the approach: After determining the speed with the 3 minute rule, you have 2 marks on your map, let's go ahead and use them for another task. Take out your ruler and draw a line that crosses those 2 marks, and there's your target track. To increase accuracy you can add some more marks before you draw the line, or add them later to verify your estimation of the target's course.
In the video I drew a line that's 9nm long on the map. The reason I did that is, that it gives me another piece of information without me having to calculate, ... a vessel running 9kts will travel a distance on 9nm in exactly 1 hour, so no I know exactly where the target will be 1 hour from now. (You have to make sure that the 9nm are measured between the target's current position and the end of the line if you chose to do it like seen in the video. Of course you can use other tools to measure and mark those 9nm on the map, that's up to you )
Now that i know where the target will be, I take out the protractor and draw a 90° angle for my approach at the 9nm point. I want to be within about 700yds of the target when I fire my torpedoes, so i make sure to plot my course accordingly. Make sure that, if you have a similar setup on the map, concerning your position relative to the target's position, that you don't get too close to the target while approaching your "hunting spot" on the surface. Better run faster a bit further away to get out of sight (I ran at flank on a parralel course until i intercepted my 90° approach line just to make sure. Of course flank isn't really necessary, but I won't get into detail on those calculations in this tutorial)
Getting the lead angle: Now we are at ~00:50s in the video, it's time to find out the lead angle for the attack. The lead angle is where your periscope will point to later, while your ship points to the impact point, the 9nm "hunting spot" in this tutorial.
What you do now is, you draw a line from the impact point towards the target. The line's length is the target's speed * 100 yards, so in our example it's 9kts * 100yds = 900yds. Then you draw a line representing the torpedo speed of 46 kts (4600yds) towards your boat. The last stept is to measure the angle between the torpedo line and the imaginary line from torpedo line end point to target speed line end point. If that sounds too cryptic, check it out in the video, I hope that will show it better.
In this tutorial with an approach angle of 90°, a target speed of 9kts and
mk14 torpedoes on high speed setting running 46kts, our lead angle is 11°.
Setting up the shot: Make sure your boat is properly aligned, so that your bow points at the impact point. What follows now is always the same process, no matter how far you are away from the target (the closer the better in terms of accuracy, therefore i chose ~700yds)
Go to the periscope, make sure your periscope is at 0° and then enter the following data into the TDC:
- AoB = 0 -> send
- Speed = 0 -> send
- Distance = max (turn the distance wheel to the right, should give you about 1200yds in the tdc after you send it) -> send
Now turn your periscope towards the target until you match the lead angle, in this example 11° to the left which means you have to go to 349° (360-11°). If the target would cross your view from right to left you would point your periscope at 11°.
Set up your torpedoes to run at high speed (46kts). You can also do this with slow running torpedoes or totally different speeds if you have some fancy modded ordnance, just use the proper yards representation when determining the lead angle.
You won't need to touch the gyro dial! Your "spread" will be archieved by the target travelling along it's course, you'll see further down.
Don't forget to open your tubes before you shoot, the time it takes to open them will mess up your solution, so better open them early.
That's it for the Setup. Leave the position keeper OFF, you won't need it.
The moment of truth: When the target is close to your lead angle - I usually monitor the target with sonar until it's about 10° away from the lead angle, so in this example I raise the scope when the target passes 339° (not sure if I did that in the video, might be a bit later there, don't get confused
).
Now all you have to do is, wait for the target to pass your scope and fire at the parts you want to hit. What I usually do is, I shoot the first torpedo as soon as the bow is in the center of the scope, then wait for the center (usually between command deck and funnel for a merchant) to fire #2 and finally i put a third one under the last 3rd of the target (usually that's the stern mast)
And that's it. If everything went right, the target should be on its way to some deep sea exploration.
//sidenote: 90° isn't a good choice for this method, especially with mk14 torpedos. Those fish tend to turn out as duds very often if you hit the target at 90°. Chosing a steeper angle reduces the dud risk, and since this method's advantage is the fact that you can choose any approach angle you want, you should use that.