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1 Chapter 8 Vector Products Revisited: A New and Eæcient Method of Proving Vector Identities Proceedings NCUR X. è1996è, Vol. II, pp. 994í998 Jeærey F. Gold Department of Mathematics, Department of Physics University of Utah Don H. Tucker Department of Mathematics University of Utah Introduction The purpose of these remarks is to introduce a variation on a theme of the scalar èinner, dotè product and establish multiplication in R n. If a =èa 1 ;:::;a n è and b =èb 1 ;:::;b n è, we deæne the product ab ç èa 1 b 1 ;:::;a n b n è. The product is the multiplication of corresponding vector components as in the scalar product; however, instead of summing the vector components, the product preservesthem in vector form. We deæne the inner sum èor traceè of a vector a =èa 1 ;:::;a n è by çèaè =a 1 + æææ+ a n. If taken together with an additional deænition of cyclic permutations of a vector hpi a ç èa 1+pè modnè ;:::;a n+pè modnè è, where a 2 R n 1
2 CHAPTER 8. VECTOR PRODUCTS REVISITED 2 and the permutation exponent p 2 Z, we are able to prove complicated vector products ècombinations of dot and cross productsè extremely eæciently, without appealing to the traditional èand cumbersomeè epsilon-ijk proofs. When applied to determinants, this method hints at the rudiments of Galois theory. Multiplication in R n DEFINITION 1 Suppose a and b 2 R n,then ab ç èa 1 b 1 ;a 2 b 2 ;:::;a n b n è : The product is the multiplication of corresponding vector components common to the scalar product, however, instead of summing the vector components, the product preserves them in vector form. THEOREM 1 If a, b, c 2 R n and æ 2 R, then Trivial. è1:1è aèbcè =èabèc ; è1:2è aèb + cè =ab + ac ; è1:3è æèabè = èæaèb = aèæbè ; è1:4è 1b = b1 = b ; where 1 ç è1; 1;:::;1è 2 R n ; è1:5è ab = ba ; è1:6è aæ = æ ; where æ ç è0; 0;:::;0è 2 R n : Inner Sums and Inner Products DEFINITION 2 THEOREM 2 The inner sum èor traceè of a vector b 2 R n is deæned as ç èbè ç nx i=1 b i = b 1 + :::+ b n : If a; b; c 2 R n and æ 2 R, then è2:1è è2:2è è2:3è è2:4è çèa + bè = çèaè + çèbè çèabè =çèbaè çècèa + bèè = çèca + cbè çèæbè =æçèbè The proofs are straightforward calculations.
3 CHAPTER 8. VECTOR PRODUCTS REVISITED 3 THEOREM 3 If a and b 2 R n,thençèabè =a æ b, wherea æ b is the familiar scalar èdotè product. çèabè =çèa 1 b 1 ;:::;a n b n è=a 1 b 1 + :::+ a n b n = a æ b. REMARKS The scalar product can be generalized for n vectors. In R 3, for example, çèab æ cè =çèac æ bè =çèbcæ aè. Each of these, expanded by using the inner product, becomes çèabcè = ab æ c = jabjjcjcosèab; cè = ac æ b = jacjjbjcosèac; bè = bc æ a = jbcjjajcosèbc; aè ; respectively. Multiplying these results together, ç 3 èabcè =jajjbjjcjjabjjacjjbcjcosèa; bcècosèb; acècosèc; abè : Now, letting c = 1, we obtain ç 3 èab1è =ç 3 èabè = p njaj 2 jbj 2 jabjcos 2 èa; bècosè1; abè : Since ç 2 èabè =jaj 2 jbj 2 cos 2 èa; bè, we ænd that an alternative representation of the inner product is given by çèabè =a æ b = p njabjcosè1; abè : This is more easily seen by the following: a æ b = 1 æ ab = j1jjabjcosè1; abè, which is equivalent to p njabjcosè1; abè. A weighted inner product can deæned by w 1 a 1 b 1 + ::: + w n a n b n, where èw 1 ;:::;w n è 2 R n are the weights. DEFINITION If a, b, and w 2 R n, where w is a weighting vector and the weights w i é 0, then the Euclidean weighted inner product of a and b is deæned as çèwabè : Note that w itself may be the product of other vectors, provided that all weights in the ænal product w are positive real numbers. PERMUTATION EXPONENTS In order to represent the cross product in terms of the new product, we deæne a vector operation that cyclically permutes the vector entries.
4 CHAPTER 8. VECTOR PRODUCTS REVISITED 4 DEFINITION 3 If b 2 R n and p 2 Z, then hpi èbè ç èb 1+pèmod nè ;b 2+pèmod nè ;:::;b n+pèmod nè è ; where hpi is the permutation exponent. The cyclic permutation makes the subscript assignment i 0! i+p èmod nè for each component b i. The modulus in the subscript of each component of b is there to insure that all subscripts i satisfy the condition 1 ç i ç n. THEOREM 4 If b 2 R n and p; q 2 Z, then è4:1è è4:2è è4:3è è4:4è è4:5è hqi è hpi bè= hp+qi èbè hqi è hpi bè= hpi è hqi bè hpi èa + bè = hpi a + hpi b hpi èabè = hpi a hpi b hpi èæbè = æ hpi b è4.1è hqi è hpi bè implies the subscript assignment i 0! i + pèmod nè followed by the assignment i 00! i 0 + qèmod nè. Since i 0 = i + pèmod nè, the subscript i 00 becomes i 00 = i + p + qèmod nè. Since pèmod nè+qèmod nè =p + qèmod nè, the assignment i 00 = i +èp + qèèmod nè is equivalent to hp+qi b. è4.2è The process is equivalent to è4.1è, except the values p and q are interchanged in the assignment i 00! i+p+qèmod nè, that is, i 00! i+q + pèmod nè, which is equivalent to hpi è hqi bè. è4.3è Note hpi èa + bè = hpi èa 1 + b 1 ;:::;a n + b n è, which in turn is equal to èa 1+pèmod nè + b 1+pèmod nè ;:::;a n+pèmod nè + b n+pèmod nè è : Now wemay write this as èa 1+pèmod nè ;:::;a n+pèmod nè è+èb 1+pèmod nè ;:::;b n+pèmod nè è ; which is equivalent to hpi a + hpi b. è4.4è Here hpi èabè = hpi èa 1 b 1 ;:::;a n b n èisequivalent to èa 1+pèmod nè b 1+pèmod nè ;:::;a n+pèmod nè b n+pèmod nè è : This, in turn, is rewritten as èa 1+pèmod nè ;:::;a n+pèmod nè èèb 1+pèmod nè ;:::;b n+pèmod nè è ; which is hpi a hpi b. è4.5è In this case, hpi èæbè = hpi èæb 1 ;:::;æb n èwhich is equivalent toæ hpi b by èæb 1+pè modnè ;:::;æb n+pè modnè è=æèb 1+pè modnè ;:::;b n+pè modnè è :
5 CHAPTER 8. VECTOR PRODUCTS REVISITED 5 THEOREM 5 If b 2 R n,thençèbè =çè h1i bè=çè h2i bè=:::= çè hn,1i bè. Since the order of the components doesn't matter, the sum remains the same for all ècyclicè permutations of the components. THEOREM 6 If a; b 2 R n and p; q; p 0 ;q 0 2 Z, thençè hpi a + hqi bè=çè hp0 i a + hq 0 i bè. çè hpi a + hqi bè = çè hpi aè+çè hqi bè = çè hp0 i aè+çè hq0 i bè = çè hp0 i a + hq0 i bè THEOREM 7 If a; b; 1 2 R n,thenèabè+ h1i èabè+:::+ hn,1i èabè =1çèabè. èabè+ h1i èabè+:::+ hn,1i èabè = èa 1 b 1 ;:::;a n b n è+èa 2 b 2 ;:::;a n b n ;a 1 b 1 è+:::+èa n b n ;a 1 b 1 ;:::;a n,1 b n,1 è = èa 1 b 1 + :::+ a n b n ;a 2 b 2 + :::+ a n b n + a 1 b 1 ;:::;a n b n + a 1 b 1 + :::+ a n,1 b n,1 è = èçèabè;çè h1i èabèè;:::;çè hn,1i èabèèè = èçèabè;:::;çèabèè = 1çèabè Cross Products THEOREM 8 If a; b 2 R 3,thena æ b = h1i a h2i b, h2i a h1i b. a æ b ç èa 2 b 3, a 3 b 2 ;a 3 b 1, a 1 b 3 ;a 1 b 2, a 2 b 1 è = èa 2 b 3 ;a 3 b 1 ;a 1 b 2 è, èa 3 b 2 ;a 1 b 3 ;a 2 b 1 è = èa 2 ;a 3 ;a 1 èèb 3 ;b 1 ;b 2 è, èa 3 ;a 1 ;a 2 èèb 2 ;b 3 ;b 1 è = h1i a h2i b, h2i a h1i b : THEOREM 9 If a; b 2 R 3,thena æ b =,b æ a. a æ b = h1i a h2i b, h2i a h1i b =,è h1i b h2i a, h2i b h1i a è =,b æ a : THEOREM 10 If a; b 2 R 3,thena æ b = h1i a h2i b, h2i a h1i b = h1i èa h1i b, h1i abè = h2i è h2i ab, a h2i bè, by Theorems 4.1, 4.3, and 4.4.
6 CHAPTER 8. VECTOR PRODUCTS REVISITED 6 Vector Identities The method of proof for the subsequent theorems is as follows: Each vector identity is rewritten in terms of the deænitions of the inner product and cross product, by Theorems 3 and 8, respectively. In the case of scalar identities, terms are permutated to isolate any desired vector in its native èun-permutatedè form, by Theorem 5. Then the newly formed terms are grouped by similar permutations. It is important to recognize cross product terms, h1i a h2i b, h2i a h1i b,or inner product terms such as h1i èacè+ h2i èacè. In the latter case, for example, one adds to this the term ac èand subtracts ac from another termè, for then one recognizes ac + h1i èacè+ h2i èacè as the inner product 1èa æ cè, according to Theorem 7. THEOREM 11 If a; b; c 2 R 3, then a æ èb æ cè =b æ èc æ aè =c æ èa æ bè. a æ èb æ cè = çèa h1i b h2i c, a h2i b h1i cè = çè h2i ab h1i c, h1i ab h2i cè = çèbè h1i c h2i a, h2i c h1i aèè = b æ èc æ aè and a æ èb æ cè = çèa h1i b h2i c, a h2i b h1i cè = çè h1i a h2i bc, h2i a h1i bcè = çècè h1i a h2i b, h2i a h1i bèè = c æ èa æ bè THEOREM 12 If a; b; c 2 R 3, then a æ b æ c = bèa æ cè, cèa æ bè. a æ èb æ cè = h1i a h2i è h1i b h2i c, h2i b h1i cè, h2i a h1i è h1i b h2i c, h2i b h1i cè = h1i ab h1i c + h2i ab h2i c, h1i a h1i bc, h2i a h2i bc = bè h1i èacè+ h2i èacèè, cè h1i èabè+ h2i èabèè + abc, abc = bèac + h1i èacè+ h2i èacèè, cèab + h1i èabè+ h2i èabèè = bèa æ cè, cèa æ bè THEOREM 13 If a; b; c; d 2 R 3,thenèaæbèæècædè =èaæcèèbædè,èaædèèbæcè. èa æ bè æ èc æ dè = çè h1i èacè h2i èbdè+ h2i èacè h1i èbdè, h1i èadè h2i èbcè, h2i èadè h1i èbcèè = çèèacè h1i èbdè+ h2i èbdèè, adè h1i èbcè+ h2i èbcèè + abcd, abcdè
7 CHAPTER 8. VECTOR PRODUCTS REVISITED 7 = çèacèbd + h1i èbdè+ h2i èbdèè, adèbc + h1i èbcè+ h2i èbcèèè = çèacèb æ dèè, çèadèb æ cèè = èb æ dèçèacè, èb æ cèçèadè = èa æ cèèb æ dè, èa æ dèèb æ cè THEOREM 14 If a; b; c; d 2 R 3, then èa æ bè æ èc æ dè =bèa æ èc æ dèè, aèb æ èc æ dèè. Let èc æ dè =e, then èa æ bè æ e = h2i ab h2i e, a h2i b h2i e, a h1i b h1i e + h1i ab h1i e = bè h1i èaeè+ h2i èaeèè, aè h1i b h1i e + h2i èbeèè + abe, abe = bèae + h1i èaeè+ h2i èaeèè, aèbe + h1i èbeè+ h2i èbeèè = bèa æ eè, aèb æ eè = bèa æ èc æ dèè, aèb æ èc æ dèè THEOREM 15 If a; b; c; d 2 R 3,thena æ èb æ èc æ dèè = bèa æ èc æ dèè, èa æ bèèc æ dè. Let èc æ dè =e, then a æ èb æ eè = h1i a h2i è h1i b h2i e, h2i b h1i eè, h2i a h1i è h1i b h2i e, h2i b h1i eè = h1i ab h1i e, h1i a h1i be, h2i a h2i be + h2i ab h2i e = bè h1i èaeè+ h2i èaeèè, eè h1i èabè+ h2i èabèè + abe, abe = bèae + h1i èaeè+ h2i èaeèè, eèab + h1i èabè+ h2i èabèè = bèa æ eè, eèa æ bè = bèa æ èc æ dèè, èa æ bèèc æ dè THEOREM 16 If a; b; c 2 R 3, then èa æ bè æ èèb æ cè æ èc æ aèè = èc æ èa æ bèè 2. First, èb æ cè æ èc æ aè = h1i è h1i b h2i c, h2i b h1i cè h2i è h1i c h2i a, h2i c h1i aè, h2i è h1i b h2i c, h2i b h1i cè h1i è h1i c h2i a, h2i c h1i aè = è h2i bc, b h2i cèèc h1i a, h1i caè,èb h1i c, h1i bcèè h2i ca, c h2i aè = h1i a h2i bcc, a h2i bc h1i c, h1i abc h2i c + ab h1i c h2i c,ab h1i c h2i c + h2i abc h1i c + a h1i bc h2i c, h2i a h1i bcc = ccè h1i a h2i b, h2i a h1i bè+c h1i cè h2i ab, a h2i bè +c h2i cèa h1i b, h1i abè = ccè h1i a h2i b, h2i a h1i bè+c h1i ècè h1i a h2i b, h2i a h1i bèè
8 CHAPTER 8. VECTOR PRODUCTS REVISITED 8 +c h2i ècè h1i a h2i b, h2i a h1i bèè = cècèa æ bèè + c h1i ècèa æ bèè + c h2i ècèa æ bèè = cèc æ èa æ bèè Then, èa æ bè æ èèb æ cè æ èc æ aèè Determinants = çèèa æ bècèc æ èa æ bèèè = èc æ èa æ bèèçècèa æ bèè = èc æ èa æ bèèèc æ èa æ bèè = èc æ èa æ bèè 2 DEFINITION 4 The P alternating vector è1;,1; 1;,1;:::èforthevector space R n n is deæned i=1 è,1èi,1^e i,wherethe^e i are orthonormal vectors. THEOREM 17 If a; 2,thendetèa; bè =çè@ a h1i bè. detèa; bè = a 1 b 2, a 2 b 1 = çèèa 1 ;a 2 èèb 2 ;,b 1 èè = çèè1;,1èèa 1 ;a 2 èèb 2 ;b 1 èè = çè@ a h1i bè THEOREM 18 If a; b; c 2 R 3, then detèa; b; cè = a æ èb æ cè = çèaèb æ cèè = çèaè h1i b h2i c, h2i b h1i cèè THEOREM 19 If a; b; c; 4, then detèa; b; c; dè = çè@ a è h1i bè h2i c h3i d, h3i c h2i dè + h2i bè h3i c h1i d, h1i c h3i dè + h3i bè h1i c h2i d, h2i c h1i dèèè
9 CHAPTER 8. VECTOR PRODUCTS REVISITED 9 THEOREM 20 If a; b; c; d; e 2 R 5,then detèa; b; c; d; eè = çèa è h1i bè h2i cè h3i d h4i e, h4i d h3i eè + h3i cè h4i d h2i e, h2i d h4i eè + h4i cè h2i d h3i e, h3i d h2i eèè + h2i bè h3i cè h1i d h4i e, h4i d h1i eè + h4i cè h3i d h1i e, h1i d h3i eè + h1i cè h4i d h3i e, h3i d h4i eèè + h3i bè h4i cè h1i d h2i e, h2i d h1i eè + h1i cè h2i d h4i e, h4i d h2i eè + h2i cè h4i d h1i e, h1i d h4i eèè + h4i bè h1i cè h3i d h2i e, h2i d h3i eè + h2i cè h1i d h3i e, h3i d h1i eè + h3i cè h2i d h1i e, h1i d h2i eèèèè According to Galois theory, roots of polynomials of degree 5 and higher cannot be characterized by closed-form solutions. Associated with each determinant is a characteristic polynomial which may reveal the connection between Galois theory and the determinant representation given by our vector product. A quick comparison of Theorems 19 and 20 reveals that the latter determinant cannot be expressed in terms of only cyclical and reverse-cyclical permutations of the permutation exponents in their natural, ascending order. With further study, we anticipate establishing the connection between the vector product representation and the group-theoretic results of Galois. Conclusion The utilization of this method for proving vector identities is reinforced by the very simple rules and notation. When applied to determinants, this method hints at the rudiments of Galois theory. Further study in this area by the authors will hopefully establish that connection in the future.
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