Vector, in Mathematics, is any directed straight line of definite length. It is conceived of most simply as a carrying or transference or step from any chosen point to that other point lying in the proper direction and at the proper distance. Take any three points ABC. Then it is evident that the operation of passing from A to C is equivalent to the successive operations of passing from A to B, and then from B to C. In other words, the vector AC is equal to the sum of the vectors AB and BC. Of course if we limit our consideration to lengths only this theorem does not hold. It is easy to see that two vectors cannot be equal unless they have the same length and the same direction. The opposite sides of a parallelogram are equal vectors. On the other hand, the radii of a sphere, although they are all of the same length, are different vectors. The fundamental law of vector addition, which has been given above, is illustrated by the law of combination of a great many important physical quantities, such as velocity, force, rotation, couple, and so on. These can all be represented by directed straight lines, and are called vector quantities (see COMPOSITION). Laws for the multiplication and division of vectors have been given by Hamilton (see QUATERNIONS) and partly by Grassmann.
For a discussion of the chief properties of vectors, see Maxwell's Matter and Motion, Clifford's Kinematic, Kelland and Tait's Quaternions.
Veda—from vid, 'to know,' hence 'knowledge, (sacred) science'—is the name of a body of religious writings which the Hindus believe to be divinely inspired. This body primarily consists of four collections (saṃhitā) of hymns, detached verses, and sacrificial formulas—viz. the Rigveda, or Veda of praises or hymns; the Sāmaveda, or Veda of chants or tunes; the Yajurveda, or Veda of prayers; and the Atharvaveda, or Veda of the Atharvans—to each of which are attached certain theological prose-works, called Brahmana, and intended chiefly to elucidate the meaning and application of the sacred texts, especially from a sacrificial point of view. The first three Vedas are often referred to as the 'trayi vidyā,' or threefold science; and they alone must originally have formed the sacred canon, whilst the fourth Veda, which is less archaic in language, was not recognised till a later period. The Samhitās of the Sāmaveda and Yajurveda are of a purely sacrificial and professional character, being intended to serve as text-books for two of the four chief classes of priests—viz. the chanters (Udgātar) and the offering priests (Adhvaryu) respectively; and the verses contained in them are to a large extent taken from the Riksamhitā, though not unfrequently with considerable textual variations. The Riksamhitā, on the other hand, though likewise assigned to a special class of priests—viz. the invokers (or sacrificers, Hotar), is not a sacrificial text-book in the same narrow sense of the word; but it has rather to be looked upon as a collection of all the sacred poetry which was within reach of the collectors, and seemed to them worthy of being preserved for devotional purposes. The hymns recited by the Hotar at the sacrifices were, however, drawn almost exclusively from this collection, the study of which was therefore especially incumbent on him. The fourth class of priests, the Brahmins, whose function it was to superintend the entire sacrificial performance, had not, and could not from the very nature of their office have, any special text-book assigned to them; and it was doubtless from mere theoretic considerations that the Atharvaveda came ultimately to be regarded as specially connected with the Brahman priest.
More or less closely connected with the Brâhmanas are two classes of works—viz. the Âraṇyakas, or forest treatises, so called from the recondite nature of their subjects, which it required an anchorite's seclusion to study adequately; and the Upanishads, or secret doctrine, treating of the nature of the Brahman, or supreme spirit. The gradual elaboration of the sacrificial system also at length gave rise to ceremonial manuals, composed in the form of strings of concise rules (sūtra). These Kalpasūtras usually consist of two parts—viz. Srautasūtras and Grihyasūtras, dealing with the great Vedic sacrifices and the ordinary domestic offerings and ceremonies respectively. In course of time the oral transmission of the canonical scriptures led to more or less serious discrepancies, as regards both the order of arrangement and textual readings, which gave rise to different schools recognising different recensions (s'ākhā, lit. 'branch') of the sacred texts. The number of such different versions was at one time very considerable, especially in the Sāmaveda and Yajurveda; but only a few of them have been preserved.
The Rigveda-saṃhitā has come down to us in a single recension, that of the S'ākala school. It consists of 1028 hymns, composed in various metres, and arranged in ten books, or mandalas. Book i. is mainly made up of sixteen collections of hymns ascribed to as many poets belonging to different families. Books ii.–vii., on the other hand, are attributed each to a special family of Rishis or seers—viz. the Gritsamadas, Kus'ikas (or Viśvāmitras), Vāmadevyas, Atris, Bharadvājas, and Vasishthas respectively—whence they are usually called the family books. In the several collections of these seven books the hymns are arranged according to the deities to whom they are addressed—viz. Agni (god of fire), Indra (god of the sky), followed by minor deities. Book viii. consists mainly of hymns adapted for chanting, and for the most part belonging to the Kāṇva family of seers. Book ix. is made up exclusively of hymns addressed to Soma, the deified power of the intoxicating juice of the soma-plant, and used for recitation at the Soma-sacrifice (see SOMA). Book x., consisting like the first of 191 hymns, has the appearance of an appendage, containing for the most part hymns of later origin, though also doubtless including not a few genuine old hymns which had probably escaped the attention of the original collectors. The date at which the hymns were collected may be approximately fixed at 1000 B.C., while the composition of the earlier hymns would probably go back several centuries before that period. The Vedic Aryans, as pictured in the hymns of the Rik, were scattered in numerous clans over the Punjab and adjoining districts west of the Indus. Their religion was mainly a worship of the powers and phenomena of nature; the favourite deities being Indra, for ever battling with the demons of darkness and drought; Agni (ignis), the genial inmate of the human dwelling in the shape of the household fire; Ushas (aurora), the fair awakener of the daily life; Soma; and Varuṇa (oḱpavos), the stern and ever-wakeful guardian of the sacred ordinances.
There is an English translation of the Riksamhitā, based on the native traditional interpretation, by H. H. Wilson, completed by E. B. Cowell; another, in metre, by Ralph Griffith; the hymns to the Maruts, or storm-gods, by F. Max-Müller.
The Brâhmana of this Veda has been handed down in two distinct versions differing considerably, especially in their arrangement of the common matter—viz. the Aitareya-brâhmana (trans. by M. Haug) of the school of the Aitareyins; and the Kauśitaki (or S'āṅkhāyana)-brâhmana of the Kauśitakins. The work deals exclusively with the various forms of Soma-sacrifice, and the consecration of the king. Each of the two schools has also its own Âraṇyaka—the Aitareyâraṇyaka and Kauśitaki-âraṇyaka—certain portions of which are styled Upanishads. Similarly there are two complete sets of ceremonial rules, or Kalpasūtras, ascribed to Âśvalâyana (? 4th or 5th century B.C.) and S'āṅkhāyana respectively.
The Sāmaveda-saṃhitā consists of two parts (or ārchika), the first of which contains the (585) verses to which the several sāman-tunes are usually sung, whilst the second gives the text of the made-up chants in the order in which they are required in the sacrificial ritual. The tunes themselves, on the other hand, are given in four special tune-books, called Gāna, attached to the Samhitā. The latter, consisting as it does mainly of detached verses taken from the Riksamhitā, has only a technical interest. The mode of chanting somewhat resembled the Gregorian or Plain chant. The Brâhmanas of this Veda have a special character of their own, inasmuch as they are not different versions, or editions, of the same traditional exegetic and legendary matter, but altogether distinct treatises, dealing chiefly with chants. Nine such works are hitherto known, the most important of which are the Tāṇḍya-brâhmana, also called Panchavims'a, consisting as it does 'of twenty-five chapters; the Shadviṃsa'-br., or 'twenty-sixth; the Chhāṇḍogya'-br., including the Chhāṇḍogya-panishad (trans. Røer, F. Max-Müller); and the Jaiminīya- or Talavakāra'-br., of which the interesting Kena-upanishad (trans. Røer; F. Max-Müller), on the nature of the Brahman, forms part. There are also several ceremonial Sūtra-works connected with this Veda, among which may be mentioned the Srautasūtra of Lātyāyana and the Grihyasūtra of Gobhila.
The Yajurveda offers the spectacle of a complete schism, its teachers and followers dividing themselves into an older and a younger branch, or the Black and the White Yajurveda, so called from the form in which their canonical books were handed down. For whilst the scriptures of the older branch presented a somewhat confused appearance, caused by the constant intermingling of the sacrificial formulas (yajus) and the exegetic portions (brâhmana), the younger school adopted the practice of the followers of the Rik by dividing their scriptures into a regular Samhitā, or collection of sacrificial formulas, and a Brâhmana. The older branch became again split up into numerous schools; four different recensions of their texts being so far known to us, the most important of which is the Taittirīya-saṃhitā, with an appendix called Taittirīya-brâhmana, though characterised by the same motley intermixture of textual and exegetic matter. There is further a Taittirīya-âraṇyaka, part of which constitutes the Taittirīya-panishad (trans. Røer; F. Max-Müller). Of the
Sūtra-works of the older schools several have come down to us, at least in part, the most complete being the Apastamba-Kalpasūtra. Of the texts of the White Yajus—viz. the Vājasaneyi-saṃhitā, and the Satapatha-brāhmaṇa, or Brāhmaṇa of a hundred paths (books i.–vii. trans. by J. Eggeling), which includes the Bṛhadāraṇyaka, the recensions of two different schools are so far known to us—viz. the Mādhyandinas and Kāṇvas, both of which recognised a common set of ceremonial rules, the Sṛautasūtra of Kātyāyana and the Kātya-Gṛhiya-sūtra of Pāraskara.
The Atharvaveda-saṃhitā is a collection of hymns and spells which, in importance and interest, ranks next to the Riksaṃhitā. The two collections present two different aspects of the religious belief of the ancient Hindus. Whilst the Rik reflects a simple belief in divine powers who are, on the whole, well disposed toward the Aryan man, and whose favour the worshipper is confident of gaining by his song, the Atharvan, on the other hand, reveals a superstitious dread of a host of malevolent powers, the effects of whose ill-will man seeks to avert by means of incantations and magic practices. As the language of the Atharvan presents a more modern appearance than the main body of the Rik, the marked change in the religious spirit of the people may perhaps be due to contact and intermixture with non-Aryan tribes. It is not impossible, however, that superstitious rites of the kind favoured in the Atharvan had long been practised by the Aryan people, though they found no expression in the devotional lyrics approved by the priests. The Saṃhitā, which is traditionally connected with the priestly families of the Atharvans and Angiras, is divided into twenty books, the last two of which are, however, later additions. The ordinary text of Northern India is usually ascribed to the Sāṇaka school. A new recension lately discovered in Kashmir not only differs from it considerably in its arrangement, but also contains some new matter, amounting to about one-sixth of the whole. The Brāhmaṇa of this Veda, the Gopatha (or cow-path)-brāhmaṇa, containing cosmogonic speculations and explanations of certain sacrificial rites, is probably a comparatively modern work; whilst of the two ceremonial sūtras, the Vaitāna (or sṛauta)-sūtra and the Kausīka-sūtra, a manual of domestic rites, the latter is by far the more interesting. The Atharvaveda has also usually assigned to it a large number of Upanishads, amounting to considerably over a hundred. They are evidently of various ages, many of them coming down to very recent times.
See the article SANSKRIT; Max-Müller, History of Ancient Sanskrit Literature; J. Muir, Original Sanskrit Texts; W. D. Whitney, essay on Veda, in Oriental and Linguistic Studies.