3. If λ, μ, v be any three non-coplanar vectors, and 4 = Γμν.φλ + Γυλ.φμ + Γλμ.φν, show that q is necessarily divisible by S.Apv. Also show that the quotient is m, — 2 €, where Vep is the non-commutative part of op. where r is a given quaternion, show that m = Σ(S.a,a,a,S.ß‚ß2ß1)+ΣS(rVa1a2.Vß2ß1)+SrΣS.aßr-Σ(SarSßr)+SrTr2, and mp-1σ = Σ(Va1a,S.ẞ2ß,0)+ΣV.aV (Vßo.r) + VorSr-VrSor. show that the following relations exist among any five qua- 0 = p(qrst)+q(rstp)+r(stpq)+s(tpqr)+t(pqrs), and q(prst) = [rst] Spq-[stp]Srq+[tpr] Ssq-[prs] Stq. S Elements, p. 492. 7. Show that if 4, be any linear and vector functions, and a, B, y rectangular unit-vectors, the vector where a, ß, y are any three vectors, then -S'. aẞy.-'p = a,Sa, p+B,Sẞ1p+Y1Syıp, 9. Show that any self-conjugate linear and vector function may in general be expressed in terms of two given ones, the expression involving terms of the second order. Show also that we may write $ + z = a(☎ + x)2 + b (w + x)(w+y)+c(w+y)2, where a, b, c, x, y, z are scalars, and and w the two given functions. 10. Solve the equations: (a.) q2 = 5qi + 10j. (b.) q2 = 2q+i. (c.) qaq=bq+c. (d.) aq= qr = rb. CHAPTER VI. GEOMETRY OF THE STRAIGHT LINE AND PLANE. 188. HAVING, in the five preceding Chapters, given a brief exposition of the theory and properties of quaternions, we intend to devote the rest of the work to examples of their practical application, commencing, of course, with the simplest curve and surface, the straight line and the plane. In this and the remaining Chapters of the work a few of the earlier examples will be wrought out in their fullest detail, with a reference to the first five whenever a transformation occurs; but, as each Chapter proceeds, superfluous steps will be gradually omitted, until in the later examples the full value of the quaternion processes is exhibited. 189. Before proceeding to the proper business of the Chapter we make a digression in order to give a few instances of applications to ordinary plane geometry. These the student may multiply indefinitely with great ease. (a.) Euclid, I. 5. Let a and ẞ be the vector sides of an isosceles triangle; ẞ-a is the base, and The proposition will evidently be proved if we show that a(a-3) Kß(B-a)-1 (§ 52). = (b.) Euclid, I. 32. Let ABC be the triangle, and let AC where y is a unit-vector perpendicular to the plane of the triangle, If = 1, the angle CAB is a right angle (§ 74). π π Hence 4=1 (§ 74). Let B = m 2' › C=n. We have This is, properly speaking, Legendre's proof; and might have been given in a far shorter form than that above. In fact we have for any three vectors, which contains Euclid's proposition as a mere particular case. (c.) Euclid, I. 35. Let B be the common vector-base of the parallelograms, a the conterminous vector-side of any one of them. For any other the vector-side is a+xß (§ 28), and the proposition appears as TVB(a+xB) TVßa (§§ 96, 98), = which is obviously true. (d.) In the base of a triangle find the point from which lines, drawn parallel to the sides and limited by them, are equal. If a, ẞ be the sides, any point in the base has the vector which bisects the vertical angle of the triangle. This is not the only solution, for we should have written instead of the less general form above which tacitly assumes that 1-x and x are positive. We leave this to the student. (e.) If perpendiculars be erected outwards at the middle points of the sides of a triangle, each being proportional to the corresponding side, the mean point of the triangle formed by their extremities coincides with that of the original triangle. Find the ratio of each perpendicular to half the corresponding side of the old triangle that the new triangle may be equilateral. Let 2a, 26, and 2 (a+B) be the vector-sides of the triangle, i a unit-vector perpendicular to its plane, e the ratio in question. The vectors of the corners of the new triangle are (taking the corner opposite to 23 as origin) |