(In geometrical terms, plot is the time or X-axis of a story, and the other axes should be Y=possession/political alignment and Z=theme, with characters being the vectors that move through this space.)
The thematic vector value of the character is found by multiplying the character's goal by the character's method. There must also exist a character with either an opposing goal or an opposing method; there may exist one character with an opposing goal and another character with an opposing method. There may additionally exist a character with both an opposing goal and an opposing method (although this would be a bit redundant). Finally, if the goal and/or method are not binary choices, there may exist multiple characters with goals and/or methods which oppose the first characters and each others'. For example, let us imagine a world with 3 types of magic: black, white, and gray. These are philosophies or methods of doing magic. Thematically, there are three possible conflicts we could choose to explore: black vs. white, black vs. gray, or white vs. gray. If we want to explore all three conflicts in the same book this gives us a nice conflict triangle: black vs. white vs. gray (vs. black again). To illustrate this conflict we need at least one character to represent each faction. (Although, keep in mind that a character does not have to be a person, and even if they are a person they might only exist off-screen, in the other characters' thoughts and dialogue.)
There are lots of ways this set-up could be developed, but I'll pick one particular way to illustrate: Let's suppose the author wants to convey that extremes are dangerous and the correct method is balance. Therefore, the gray mage is the 'good guy' and the white and black mages are the 'bad guys'. But, the gray mage is not necessarily the viewpoint character; if he started out already having the right method, what would there be for him to learn during the story?
- Perhaps the viewpoint character is a student mage who has not picked his color yet, and the gray mage is his positive mentor or guardian.
- Or perhaps the viewpoint character is the white mage who is friends with the gray mage and enemies with the black mage. The white mage would always be arguing with his friend about whether white magic or gray magic was better, and at the end of the story the white mage might decide gray magic is better and change his color to match that of his friend.
- Or, perhaps the viewpoint character is the gray mage. He is in love with the black mage, but cannot marry her because they belong to opposing factions. So they argue throughout the story and at the end the black mage realizes grayness is better and changes to become a gray mage and they live happily ever after.
- Or perhaps the viewpoint character gray mage is torn between being in love with twins, one of whom is a black mage and one of whom is a white mage, and who are always fighting with each other. The twins have a climactic fight and accidentally get magically merged into one body, averaging their blackness and whiteness into grayness.
- Or perhaps there is no gray mage; the black mage and the white mage start out as enemies, fall in love, and relize that when they work together their magic averages out to gray.
So there are many possibilities, but the point is that the author picks a theme (black/white/grayness) and a moral argument to make within this theme (i.e. grayness is correct and blackness and whiteness are both incorrect), creates characters to represent the various sides of the argument, argues the point through the conflict between the characters, and shows the superiority of one side by neutralizing the characters representing the other sides. A character can be neutralized by being killed, by being shown to be incompetent and thus harmless, by being permanently locked in a struggle with an equal and opposite character such that they have no time to bother anyone else, or by being convinced to change and join the 'correct' side.