Vector Products Revisited: A New and Efficient Method of Proving Vector Identities
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This document presents a new method for proving vector identities using a variation of the scalar dot product and cyclic permutations in n-dimensional space. It introduces the concept of preserving vector components in their product form rather than summing them and offers theorems and definitions related to vector multiplicative operations, inner products, and cross products. The approach significantly simplifies the proof of complex vector identities, hinting at connections to Galois theory.
Vector Products Revisited: A New and Efficient Method of Proving Vector Identities
- 1. Chapter 8 Vector Products Revisited: A New and E cient Method of Proving Vector Identities Proceedings|NCUR X. 1996, Vol. II, pp. 994 998 Je rey 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 inner, dot product and establish multiplication in Rn . If a = a1 ; : : : ; an and b = b1 ; : : : ; bn, we de ne the product ab a1b1; : : : ; anbn. The product is the multiplication of corresponding vector components as in the scalar product; however, instead of summing the vector components, the product preserves them in vector form. We de ne the inner sum or trace of a vector a = a1 ; : : : ; an by a = a1 + + an. If taken together with an additional de nition of cyclic permutations of a vector hpi a a1+p mod n ; : : : ; an+p mod n , where a 2 Rn 1
- 2. CHAPTER 8. VECTOR PRODUCTS REVISITED 2 and the permutation exponent p 2 Z, we are able to prove complicated vector products combinations of dot and cross products extremely e ciently, without appealing to the traditional and cumbersome epsilon-ijk proofs. When applied to determinants, this method hints at the rudiments of Galois theory. Multiplication in Rn DEFINITION 1 Suppose a and b 2 Rn , then ab a1b1; a2b2; : : : ; anbn : 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 Rn and 2 R, then 1:1 abc = abc ; 1:2 ab + c = ab + ac ; 1:3 ab = ab = a b ; 1:4 1b = b1 = b ; where 1 1; 1; : : : ; 1 2 Rn ; 1:5 ab = ba ; 1:6 a = ; where 0; 0; : : : ; 0 2 Rn : Proof: Trivial. Inner Sums and Inner Products DEFINITION 2 The inner sum or trace of a vector b 2 Rn is de ned as b Xb = b n i 1 + : : : + bn : i=1 THEOREM 2 If a; b; c 2 Rn and 2 R, then 2:1 a + b = a + b 2:2 ab = ba 2:3 ca + b = ca + cb 2:4 b = b Proof: The proofs are straightforward calculations.
- 3. CHAPTER 8. VECTOR PRODUCTS REVISITED 3 THEOREM 3 If a and b 2 Rn , then ab = a b, where a b is the familiar scalar dot product. Proof: ab = a1 b1 ; : : : ; an bn = a1 b1 + : : : + an bn = a b. REMARKS The scalar product can be generalized for n vectors. In R3, for example, ab c = ac b = bc a. Each of these, expanded by using the inner product, becomes abc = ab c = jabjjcjcosab; c = ac b = jacjjbjcosac; b = bc a = jbcjjajcosbc; a ; respectively. Multiplying these results together, 3 abc = jajjbjjcjjabjjacjjbcjcosa; bccosb; accosc; ab : Now, letting c = 1, we obtain 3 ab1 = 3 ab = pnjaj2 jbj2 jabjcos2 a; bcos1; ab : Since 2 ab = jaj2 jbj2 cos2 a; b, we nd that an alternative representation of the inner product is given by p ab = a b = njabjcos1; ab : This is more easily seen by the following: p a b = 1 ab = j1jjabjcos1; ab, which is equivalent to njabjcos1; ab. A weighted inner product can de ned by w1 a1 b1 + : : : + wn an bn , where w1 ; : : : ; wn 2 Rn are the weights. DEFINITION If a, b, and w 2 Rn , where w is a weighting vector and the weights wi 0, then the Euclidean weighted inner product of a and b is de ned as wab : Note that w itself may be the product of other vectors, provided that all weights in the nal product w are positive real numbers. PERMUTATION EXPONENTS In order to represent the cross product in terms of the new product, we de ne a vector operation that cyclically permutes the vector entries.
- 4. CHAPTER 8. VECTOR PRODUCTS REVISITED 4 DEFINITION 3 If b 2 Rn and p 2 Z, then hpi b b1+pmod n ; b2+pmod n ; : : : ; bn+pmod n ; where hpi is the permutation exponent. The cyclic permutation makes the sub- script assignment i0 ! i + p mod n for each component bi . The modulus in the subscript of each component of b is there to insure that all subscripts i satisfy the condition 1 i n. THEOREM 4 If b 2 Rn and p; q 2 Z, then 4:1 hqi hpi b = hp+qi b 4:2 hqi hpi b = hpi hqi b 4:3 hpi a + b = hpi a + hpi b 4:4 hpi ab = hpi a hpi b 4:5 hpi b = hpi b Proof: 4.1 hqi hpi b implies the subscript assignment i0 ! i + pmod n followed by the assignment i00 ! i0 + qmod n. Since i0 = i + pmod n, the subscript i00 becomes i00 = i + p + qmod n. Since pmod n + qmod n = p + qmod n, the assignment i00 = i + p + qmod n is equivalent to hp+qi b. 4.2 The process is equivalent to 4.1, except the values p and q are inter- changed in the assignment i00 ! i + p + qmod n, that is, i00 ! i + q + pmod n, which is equivalent to hpi hqi b. 4.3 Note hpi a + b = hpi a1 + b1 ; : : : ; an + bn, which in turn is equal to a1+pmod n + b1+pmod n ; : : : ; an+pmod n + bn+pmod n : Now we may write this as a1+pmod n ; : : : ; an+pmod n + b1+pmod n ; : : : ; bn+pmod n ; which is equivalent to hpi a + hpi b. 4.4 Here hpi ab = hpi a1 b1 ; : : : ; an bn is equivalent to a1+pmod n b1+pmod n ; : : : ; an+pmod n bn+pmod n : This, in turn, is rewritten as a1+pmod n ; : : : ; an+pmod n b1+pmod n ; : : : ; bn+pmod n ; which is hpi a hpi b. 4.5 In this case, hpi b = hpi b1 ; : : : ; bn which is equivalent to hpi b by b1+p mod n ; : : : ; bn+p mod n = b1+p mod n ; : : : ; bn+p mod n :
- 5. CHAPTER 8. VECTOR PRODUCTS REVISITED 5 THEOREM 5 If b 2 Rn, then b = h1i b = h2i b = : : : = hn,1i b. Proof: Since the order of the components doesn't matter, the sum remains the same for all cyclic permutations of the components. THEOREM 6 If a; b 2 Rn and p; q; p0; q0 2 Z, then hpi a + hqi b = hp i a + 0 hq i b. 0 Proof: hpi a + hqi b = hpi a + hqi b = hp i a + hq i b 0 0 = hp i a + hq i b 0 0 THEOREM 7 If a; b; 1 2 Rn , then ab + h1i ab + : : : + hn,1i ab = 1 ab. Proof: ab + h1i ab + : : : + hn,1i ab = a1 b1 ; : : : ; an bn + a2 b2; : : : ; an bn ; a1 b1 + : : : + an bn ; a1 b1 ; : : : ; an,1 bn,1 = a1 b1 + : : : + an bn ; a2 b2 + : : : + an bn + a1 b1 ; : : : ; an bn + a1 b1 + : : : + an,1 bn,1 = ab; h1i ab; : : : ; hn,1i ab = ab; : : : ; ab = 1 ab Cross Products THEOREM 8 If a; b 2 R3 , then a b = h1i ah2i b , h2i ah1i b. Proof: a b a2 b3 , a3b2; a3b1 , a1b3; a1b2 , a2b1 = a2 b3 ; a3 b1 ; a1 b2 , a3 b2 ; a1 b3; a2 b1 = a2 ; a3 ; a1 b3 ; b1 ; b2 , a3 ; a1 ; a2 b2 ; b3; b1 = h1i ah2i b , h2i ah1i b : THEOREM 9 If a; b 2 R3 , then a b = ,b a. Proof: ab = h1i ah2i b , h2i ah1i b = , h1i bh2i a , h2i bh1i a = ,b a : THEOREM 10 If a; b 2 R3 , then a b = h1i ah2i b , h2i ah1i b = h1i ah1i b , h1i ab = h2i h2i ab , ah2i b, 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 de nitions 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 un-permutated form, by Theorem 5. Then the newly formed terms are grouped by similar permuta- tions. It is important to recognize cross product terms, h1i ah2i b , h2i ah1i b, or inner product terms such as h1i ac + h2i ac. In the latter case, for example, one adds to this the term ac and subtracts ac from another term, for then one recognizes ac + h1i ac + h2i ac as the inner product 1a c, according to Theorem 7. THEOREM 11 If a; b; c 2 R3 , then a b c = b c a = c a b. Proof: a b c = ah1i bh2i c , ah2i bh1i c = h2i abh1i c , h1iabh2i c = bh1i ch2i a , h2i ch1i a = b c a and a b c = ah1i bh2i c , ah2i bh1i c = h1i ah2i bc , h2i ah1i bc = ch1i ah2i b , h2iah1i b = c a b THEOREM 12 If a; b; c 2 R3 , then a b c = ba c , ca b. Proof: a b c = h1i ah2i h1i bh2i c , h2i bh1i c , h2i ah1i h1i bh2i c , h2i bh1i c = h1i abh1i c + h2i abh2i c , h1i ah1i bc , h2iah2i bc = bh1i ac + h2i ac , ch1i ab + h2i ab + abc , abc = bac + h1i ac + h2i ac , cab + h1i ab + h2i ab = ba c , ca b THEOREM 13 If a; b; c; d 2 R3 , then abcd = acbd,adbc. Proof: a b c d = h1i ach2i bd + h2i ach1i bd , h1i adh2i bc , h2i adh1i bc = ach1i bd + h2i bd , adh1i bc + h2i bc + abcd , abcd
- 7. CHAPTER 8. VECTOR PRODUCTS REVISITED 7 = acbd + h1i bd + h2i bd , adbc + h1i bc + h2i bc = acb d , adb c = b d ac , b c ad = a cb d , a db c THEOREM 14 If a; b; c; d 2 R3 , then a b c d = ba c d , ab c d. Proof: Let c d = e, then a b e = h2i abh2i e , ah2i bh2i e , ah1i bh1i e + h1i abh1i e = bh1i ae + h2i ae , ah1i bh1i e + h2i be + abe , abe = bae + h1i ae + h2i ae , abe + h1i be + h2i be = ba e , ab e = ba c d , ab c d THEOREM 15 If a; b; c; d 2 R3 , then a b c d = ba c d , a bc d. Proof: Let c d = e, then a b e = h1i ah2i h1i bh2i e , h2i bh1i e , h2i ah1i h1i bh2i e , h2i bh1i e = h1i abh1i e , h1i ah1i be , h2i ah2i be + h2i abh2i e = bh1i ae + h2i ae , eh1i ab + h2iab + abe , abe = bae + h1i ae + h2i ae , eab + h1i ab + h2i ab = ba e , ea b = ba c d , a bc d THEOREM 16 If a; b; c 2 R3 , then a b b c c a = c a b2 . Proof: First, b c c a = h1i h1i bh2i c , h2i bh1i ch2i h1i ch2i a , h2i ch1i a , h2i h1i bh2i c , h2i bh1i ch1i h1i ch2i a , h2i ch1i a = h2i bc , bh2i cch1i a , h1i ca ,bh1i c , h1i bch2i ca , ch2i a = h1i ah2i bcc , ah2i bch1i c , h1i abch2i c + abh1i ch2i c ,abh1i ch2i c + h2i abch1i c + ah1i bch2i c , h2i ah1i bcc = cch1i ah2i b , h2i ah1i b + ch1i ch2i ab , ah2i b +ch2i cah1i b , h1i ab = cch1i ah2i b , h2i ah1i b + ch1i ch1i ah2i b ,h2i ah1i b
- 8. CHAPTER 8. VECTOR PRODUCTS REVISITED 8 +ch2i ch1i ah2i b , h2i ah1i b = cca b + ch1i ca b + ch2i ca b = cc a b Then, a b b c c a = a bcc a b = c a b ca b = c a bc a b = c a b2 Determinants Pi DEFINITION 4 The alternating vector 1; ,1; 1; ,1; : : : for the vector space Rn is de ned as @ n=1 ,1i,1 ei , where the ei are orthonormal vectors. ^ ^ THEOREM 17 If a; b; @ 2 R2 , then deta; b = @ ah1i b. Proof: deta; b = a1 b2 , a2 b1 = a1 ; a2 b2 ; ,b1 = 1; ,1a1 ; a2 b2 ; b1 = @ ah1i b THEOREM 18 If a; b; c 2 R3 , then Proof: deta; b; c = a b c = ab c = ah1i bh2i c , h2i bh1i c THEOREM 19 If a; b; c; d; @ 2 R4, then deta; b; c; d = @ a h1i bh2i ch3i d , h3i ch2i d + h2ibh3i ch1i d , h1i ch3i d + h3ibh1i ch2i d , h2i ch1i d
- 9. CHAPTER 8. VECTOR PRODUCTS REVISITED 9 THEOREM 20 If a; b; c; d; e 2 R5, then deta; b; c; d; e = a h1i bh2i ch3i dh4i e , h4i dh3i e + h3i ch4i dh2i e , h2i dh4i e + h4i ch2i dh3i e , h3i dh2i e + h2i bh3i ch1i dh4i e , h4i dh1i e + h4i ch3i dh1i e , h1i dh3i e + h1i ch4i dh3i e , h3i dh4i e + h3i bh4i ch1i dh2i e , h2i dh1i e + h1i ch2i dh4i e , h4i dh2i e + h2i ch4i dh1i e , h1i dh4i e + h4i bh1i ch3i dh2i e , h2i dh3i e + h2i ch1i dh3i e , h3i dh1i e + h3i ch2i dh1i e , h1i dh2i e According to Galois theory, roots of polynomials of degree 5 and higher cannot be characterized by closed-form solutions. Associated with each deter- minant 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.