へ行(い)くTo go to, To head toward

The verb いく expresses movement in a direction, literally or figuratively. へいく is used after nouns to indicate moving toward a place (e.g., 学校へいく = 'go to school'). When used as ていく (Verb + て + いく), it implies a continuing action or change moving forward. Note that いく is written in kanji as 行く when used alone, but typically in hiragana in the ていく form.

へいく is used with nouns, typically physical locations or directions, to express 'going' or 'heading' somewhere (e.g., 学校へいく, 西へいく). へ can be replaced with に, which is more colloquial and emphasizes the final destination. にいく can also combine with verb stems to express going to perform an action (e.g., 食べにいく = 'go to eat'). The two patterns can be combined as well (e.g., 店へ食べにいく = 'go to the store to eat').

The へ in both patterns is the same directional particle. While it's commonly used with いく to indicate going somewhere or moving in a direction (e.g., 学校へいく, 西へいく), it can also be used with other verbs (e.g., 家へ帰る, 公園へ歩く) or in certain phrases (e.g., 東京へようこそ！). In all cases, へ expresses directionality or movement toward a location.

くる and いく are both movement verbs often used with directional particles to express moving toward a particular location. The difference lies in the perspective of the speaker and directionality: くる typically expresses movement toward the speaker's location, while いく expresses movement away from the speaker's location.

Both いらっしゃる and いく express movement away from the speaker, similar to the English 'go.' The key difference is politeness: いく is a neutral verb used in everyday contexts, while いらっしゃる is a polite, honorific verb used to show respect for someone's movement. Note that いらっしゃる also serves as the honorific form of いる, くる, and いく.