Vyūha means - 'to arrange troops in a battle array', 'to arrange, put or place in order, to dispose, separate, divide, alter, transpose, disarrange, resolve (vowels syllables etc.)'. Its root is व्यः which means - a 'cover' or 'veil'. This word also refers to emanation and to the manifest power of Lord Vishnu.
In the Upanishads the word vyuha occurs once, it appears in śloka 16 of the Isha Upanishad :-
"O Sun, sole traveler of the Heavens, controller of all, Surya, son of Prajapati; remove thy rays and gather up the burning light. I behold thy glorious form; I am he, the Purusha within thee."
In this passage vyūha means – 'remove' (to a distance thy rays). The sage declares that the Truth is concealed in the Vedas, covered by a golden lid or vessel Badarayana, by declaring – Owing to the impossibility of origin - Brahma Sutras (II.ii.42) refutes the Bhagavata view that the Chatur-vyūha forms originate successively from Vasudeva, for any origin for the soul is impossible, an implement cannot originate from its agent who wields it. Whereas in a vyūha an army re-sets its different able warriors and weaponry into a specific arrangement as per battle demands, the Supreme Being re-sets the contents of consciousness through yogamaya with each formation concealing yet another formation. The five layers of matter (prakrti) that constitute the human body are the five sheaths (panchakosa), one moves inwards from the visible layers through more refined invisible layers in search of own true self.
The Pāñcarātra Āgama, which are based on Ekāyana recension of the Śukla Yajuveda, is later than the Vedas but earlier than the Mahabharata, the main āgamas are- the Vaiśnava (worship of Vishnu), the Śaiva (worship of Shiva) and the Śākta (worship of Devi or Shakti) āgamas; all āgamas are elaborate systems of Vedic knowledge. According to Vedanta Desika, the Pāñcarātra āgama teaches the five-fold daily religious duty consisting of – abhigamana, upādāna, ijyā, svādhyāya and yoga, the name of this āgama is derived on account of its description of the five-fold manifestation of the Supreme Being viz, para (supreme or the transcendental form), vyūha (formation or manifestation as the four vyūha), vibhava (reincarnation or descent to earth as avtāra), arcā (the visible image of God) and antaryāmi (the cosmic form of God as existing everywhere and in everything). Lakshmi accompanies Vishnu in His vyūha manifestation which is four-fold (Chatur-vyūha) – Vāsudeva (as the creator), Samkarśana (as the sustainer), Pradyumna (as destroyer - dissolution of the universe) and Aniruddha (as promulgator of spiritual knowledge). This is the Vaiśnava doctrine of Vyūha or the doctrine of formation.
The Chatur-vyūha forms of Vishnu are related to four of the six causes of creation which six are God Himself as the final cause of creation and His five aspects – Narāyana ('thinking'), Vāsudeva ('feeling'), Samkarśana ('willing'), Pradyumna ('knowing') and Aniruddha ('acting') successively; each divinity controls its specific creative energy. The six gunas – jnana (omniscience), aishvarya (lordship), shakti (potency), bala (force), virya (virtue) and tejas (self-sufficiency), acting in pairs and in totality, are the instruments and the subtle material of pure creation. Vyūhas are the first beings created, and they represent the effective parts of a coherent whole. Here, vyūha means – projection; the projection of the svarūpa ('own form') as bahurūpa ('manifest variously').
One of the five Dhyāni Buddhas, Amitābha, who does not appear in any Pali Buddhist text or in Sanskrit Buddhist text, and his active manifestation in Bodhisttava form, Avalokiteśvara, are known through the two-versioned Mahāyāna Buddhist sūtra Sukhāvatī-vyūha ('Array of the Land of Bliss') which text pertains to Gautama Buddha’s recollection of one his previous birth as Amitābha, as told to Ānanda, who was also told about 81 Buddhas of the past, the last named being Lokeśvararāja who taught Dharmākara who having practiced the virtues of a Bodhisttava and attained enlightenment was reborn as Amitābha whose land this text describes. According to Mahāyāna Buddhism, the word vyūha means – 'arrangement', the like of marvellous, supernatural, magical arrangements, or supernatural manifestations.
The Mahabharata and the Manu Samhita list by name and formation many vyūhas ('battle formations') such as shakata-vyūha ('cart-shaped formation'), garbha-vyūha ('womb-shaped formation'), suchi-vyūha ('needle-shaped formation'), ardha-chandra-vyūha ('crescent moon formation'), sarvatobhadra-vyūha ('grand formation'), makara-vyuha ('crocodile formation'), shyena-vyuha ('eagle formation') etc., some were small in size and others, gigantic. Vajra-vyūha was a three-fold formation of warriors. Chakra-vyūha ('circular formation') was devised by the Kauravas in which Abhimanyu, son of Arjuna, was trapped never to emerge alive.