Proceedings - NCUR X. (1996), Vol. II, pp. 994-998 Jeffrey 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 R^{n}. If a = (a_{1},...,a_{n}) and b = (b_{1},...,b_{n}), we define the product ab = (a_{1}b_{1},...,a_{n}b_{n }). 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 define the inner sum (or trace) of a vector a = (a_{1},...,a_{n}) by (a) = a_{1}+...+a_{n}. If taken together with an additional definition of cyclic permutations of vectors, we are able to prove complicated vector products (combinations of dot and cross products) extremely efficiently, without appealing to the traditional (and cumbersome) epsilon-ijk proofs. When applied to determinants, this method hints at the rudiments of Galois theory.

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.