Advanced Calculus fi..

Chapter 1 Vectors and Matrices 53as In relativity a 6dimensional space is needed. In quantum mechanics, one isin fact forced to consider the limiting case n = m; surprisingly enough, this theoryof a vector space of infinite dimension turns out to be closely related to the theoryof Fourier series (Chapter 7).PROBLEMS1. In v4,1etu= (3,2, 1,0),v = (1.0, 1.2),w = (5-4, I, -2).a) Find u + v, u + w, 2u, -3v, Ow.b) Find 3u - 2v, 2v + 3w, u - w, u + v - 2w.C) Find u . v, u . w, (uJ, Ivl.d) Show that w can be expressed as a lincar combination of u and v and hence that u,v, w are linearly dependent.2. In En the* segment PI P2 joining points PI. PZ is formed of all points P such thatPIP= t ~1~2,whereo~ t 5 1.a) In E4, show that (5, 10, - 1, 8) is on the line segment from (1,2,3,4) 10 (7, 14, -3, 10).b) In E4, is (2, 8, I, 6) on the line segment joining (I, 7, 2, 5) to (9,2, 0, 7)?C) In E5, find the midpoint of the line segment from PI: (2,0, l,3,7) to P2: (lo, 4, -3,3, 5); that is, find the point P on the segment such that IQI = ~ q l .d) In E5, trisect the line segment joining (7, 1, 7, 9.6) to (16, -2, 10, 3, 0).-- -__fI = I PI P2 I.-. 2-3. Prove the P5gorean theorem in En : If PI P2 is orthogonal to P2 P3, then I PI P3 1 -e) In En, show that if P is on the line segment PI P2, then I$ I + ISI PI PZl 2+ I P2 P312. [Hint: Write the left side as- f4. Prove thc Law of Cosines in En: If PI P2 and PI P3 are nonzero vectors, then' +where 8 is the angle between PI Pr and PI P3.5. a) In E5, find the sides and angles of the triangle with vertices(1,2,3,4,5), (5,4,2,3,1), and (2,2,2,2,2).b) Prove: In En, the sum of the angles of a triangle is n. [Hint: For a triangle with sidesa, b, c in En, construct the triangle with the same sides in the plane E2; why is thispossible? Then show, with the aid of thc Law of Cosines, that corresponding anglesof the two triangles are equal.]6. Prove the rules (I. 102), with the aid of (1.96) through (1.101).7. Prove Cauchy's inequality:[Hint: Use ( 1.107).]