CHAPTER VI. APPLICATIONS OF VECTOR FORMULAE 224. THE definitions of the products of vectors were in 117 expressed in trigonometrical terms. That the introduction of trigonometry was unnecessary is abundantly proved by our not having made any use of these definitions. Vector-algebra however contains implicitly the whole of trigonometry, and it is now proposed to shew how the fundamental formulae of that subject can be easily obtained from theorems in vectors. The advantage of a short account of the trigonometrical applications of vectors is threefold; (1) it gives short and easy proofs in trigonometry, (2) it is a good exercise in vector formulae and manipulation, (3) it enables us pass from general investigations to numerical calculation by the aid of trigonometrical tables. 225. Sine and Cosine. Let a and B be any two vectors of lengths a and b respectively, and let e denote the angle that a must be turned through contra-clockwise in order that it may have the same direction and sense as B, then by § 117 (a) ab cos = [aẞ] = ab sin 0. e, (where e is an ort perpendicular to the plane of the vectors, having a sense such that a, B, e form a righthanded system). If a and B are orts, then a = b = 1 and which constitute the vector definitions of the cosine and sine of the angle 0. A Note. To avoid the introduction of another symbol the angle as above defined is sometimes written aß. 226. The other trigonometrical ratios are defined from the sine and cosine, e.g. tangent = sin &c. 227. The signs of the cosine and sine given by the definition of § 225 agree with the usual ones for all values of 0. This should be tested by drawing figures for the four quadrants, remembering the rules for determining the signs of (a) and [aß]. 228. Negative Angles. By § 225 (aß) means the cosine of the angle turned through by revolving a contraclockwise till it coincides with B, hence (Ba) means the cosine of the angle turned through by revolving ß clockwise till it coincides with a. Now hence Also hence 229. Turning B through the angle π, i.e. two right angles, in either the positive or negative sense changes the sense of ß, but 230. If a and B are identical orts, then = cos 2π = COS π- - 1, sin π = 0. If a and B are perpendicular orts, then 231. Let a and B be any two orts and the angle Multiply (1) by a, taking vector products, then .(ii). ..(iii). In an exactly similar manner the values of sin and sin (+) may be obtained. 2 From (2), (3) and (4) we obtain at once [aß] = [yB] (ay) + [ay] (By) hence sin (A + B) = sin B cos A+ cos B sin A. The other fundamental formulae may be obtained from this in the usual manner or derived direct from (1) by vector manipulation. 233. The sum or difference of two sines or cosines can also easily be expressed as a product directly by means of vectors. Hence 2 (YB) [aẞ] = [ay]+[ad], which in trigonometrical form is 2 cos B sin Asin (A + B) + sin (A – B). Again or in trigonometrical form cos (A + B) + cos (A — B) : = 2 cos A. cos B. By expressing the fact that the magnitude of y-d is twice that of [By] the corresponding difference formulae are easily obtained. 234. The expansions obtained in §§ 232 and 233 are special cases of two general theorems which we now proceed to prove. Let a, B, y, 8 be any four coplanar vectors, then between any three of them, say a, B, y, there exists a relation of the form Now [By], [ya] &c. are all like vectors, and we may form vector or scalar products with these and any other vectors we please. Multiply (i) by [ad] and (ii) by [68] taking scalar products, then ([By]|[ad]) + ([ya]|[B8]) = ([Ba]|A [ad] + B [ßd]) Hence = = ([Ba]|[Aa + BB|8]) = = ([Ba]|[yd]). ([By]|[ad]) + ([ya]|[B8]) + ([aß]|[yd]) = 0................(iii). Multiplying (i) and (ii) by the scalars (ad) and (B8) we obtain in a similar manner [By] (ad) + [ya] (BS) + [aß] (yd) = 0................(iv). |