私は彼らに向かって手を振った。

I; me; private affairs; personal matter; secrecy; selfishness

topic marker particle; indicates contrast with another option (stated or unstated); adds emphasis

they (usually male); them

at (place, time); in; on; during; to (direction, state); toward; into; for (purpose); because of (reason); for; with; by; from; as (i.e. in the role of); per; in; for; a (e.g. "once a month"); and; in addition to; if; although

to face; to go towards

hand; arm; forepaw; foreleg; handle; hand; worker; help; trouble; care; effort; means; way; trick; move; technique; workmanship; hand; handwriting; kind; type; sort; one's hands; one's possession; ability to cope; hand (of cards); direction; move (in go, shogi, etc.)

indicates direct object of action; indicates subject of causative expression; indicates an area traversed; indicates time (period) over which action takes place; indicates point of departure or separation of action; indicates object of desire, like, hate, etc.

to wave; to shake; to swing; to sprinkle; to throw (dice); to cast (actor); to allocate (work); to turn down (somebody); to reject; to jilt; to dump; to abandon; to give up; to ruin; to add kana indicating a reading of a word; to slightly change headings; to change directions; to extract by broiling; to prepare an infusion of; to decoct; to carry with great vigor (e.g. a portable shrine); to bring up the main topic; to replace; to substitute; to set up a joke for somebody else