Gilmoy: <sevenseaman: Say b5-f1 is one of the lines of force of the Black Q. What all does it do and which other friendly lines of force does it intersect?>
It depends where your other pieces are. (Really, that's the point.) It's a way to think holistically, to maximize the benefit of teamwork between 2+ pieces.
"Lines of force" is simply the reachable squares of each of your pieces. Where two lines intersect, that means either of your pieces would be protected by the other one if you moved it there. That, in turn, gives you mobility to advance a plan, or shop for tactics that are nastier than your opponent's.
He who gets forward first often gets an advantage in space or initiative. In the newbie form, chess has many 1-piece maxims like <rusty nail in the kNe6> or <rook on the 7th>, or conversely <best defender of a castled K>. At intermediate levels, we decompose those advantages into multi-move plans that can force them on an unwilling opponent. "Intersections" are a key mechanism, because they can build toward winning the tempo to get the first piece that deep.
Learning the game 1 piece at a time, we quickly grasp the idea: <put it on its strongest square>, which generally means controlling the most squares: R on (half-)open file, Bs on long diagonals or pinning f7/f2, Ns not-rimward. To improve, we progress to thinking multiple pieces at a time, and go for positions with criss-crosses.
The strongest "criss-cross" is 2 line movers on the same line: we call that a "battery", and it's well-known as "when in doubt, double your rooks", or Q-B vs Dragon-like anything. Later, we grok that it would be more of a (mating) threat to have the Q in front, ergo it can be wise to invest a move to get your Bs behind -- which becomes the <GM bishop thing>, e.g. Bd3-Bb1.
The epitome of this idea, and very rare OTB, is the battery fork, or "double-double":
- A. (0.5 stars) Half-Star Bob has two en prise, Alice's Q forks them.
- B. (1.0 star) One-Star Bob has 2 pieces, both 1-defended. Alice is painting them with 1 dot (of control/attack) each. It's a wash.
- E. (5.5 stars) Alice moves her Q onto <both lines> simultaneously. Now she has 2 dots on each, and that's the same fork. It's much harder to pull off, since it presumes both a rook lift <and> a bishop thing, and that those were the two best (or at least very good) single-piece moves at those times.
Rubinstein vs Hromadka, 1923
Ivanchuk vs Vallejo-Pons, 2011
It follows that the weakest criss-cross (that you can still plan for) is B+N, since they intersect in at most 2 squares, and only every other N move. (N+N is even rarer, so you basically cannot plan to achieve it -- although Gerard Welling tries with his goofball Nf3-Nd2-Nf1-Ne3 for a double-Nd5 :) But because of that, B+N coordination becomes critically important in all openings, and separates the noobs from the tyros.
Working backward, this drives a great deal of opening theory, esp. Nc3-Bg2-Nd5, Colle System's overprotect-and-erupt on e4, Nimzo-Indian Nf6-Bb4-Ne4 to not let White do what he wants to do, Shirov's wacky g4 "sac" vs Sicilian K-side castling (and why nobody ever takes that pawn), and Nimzo-Larsen Attack 1.b3 promising a heavy dogpile on g7. All of these exploit criss-crosses to control (or blow up) the center, sometimes with an eye toward midgame attack potential.
Then since everybody just memorizes openings, good players further stratify by finding midgame repostings of B+N. Carlsen strikes me as particularly obnoxious at this (or maybe it's just book in his English/QID lines :) Fischer's <10..Ba6 14..Nd3> is a prime example: as a bottleneck, it thoroughly stymies White's Q, winning a tempo to play a precisely-calculated line. (In other words, he didn't do it just to do it, since objectively he's losing B+N for R -- but it gave him a springboard for tactics, and he saw it all the way to a winning attack.)
<How do we use these intersections?> Generally, you plant a piece where your opponent can't take it (yet). Then your piece sits in his backfield and hurts something for 1 tempo, and as he puts out fire on board, board burns faster.
Mnemonic: Every protected piece was an intersection before it got there!