3.6 Vector Identities And Notation
To begin with consider u ×
and it is desired to simplify this quantity. It turns out this is an
important quantity which comes up in many different contexts. Let
Next consider u×
which is given by
When you multiply this out, you get
and if you are clever, you see right away that
A related formula is
This derivation is simply wretched and it does nothing for other identities which may arise in applications.
Actually, the above two formulas, 3.32
are sufficient for most applications if you are creative in
using them, but there is another way. This other way allows you to discover such vector identities as the
above without any creativity or any cleverness. Therefore, it is far superior to the above nasty
computation. It is a vector identity discovering machine and it is this which is the main topic in what
There are two special symbols, δij and εijk which are very useful in dealing with vector identities. To
begin with, here is the definition of these symbols.
Definition 3.6.1 The symbol δij, called the Kroneker delta symbol is defined as follows.
With the Kroneker symbol i and j can equal any integer in
for any n ∈ ℕ.
Definition 3.6.2 For i,j, and k integers in the set,
, εijk is defined as follows.
The subscripts ijk and ij in the above are called indices. A single one is called an index. This symbol εijk is
also called the permutation symbol.
The way to think of εijk is that ε123 = 1 and if you switch any two of the numbers in the list i,j,k, it
changes the sign. Thus εijk = −εjik and εijk = −εkji etc. You should check that this rule reduces to the
above definition. For example, it immediately implies that if there is a repeated index, the answer is zero.
This follows because εiij = −εiij and so εiij = 0.
It is useful to use the Einstein summation convention when dealing with these symbols. Simply stated,
the convention is that you sum over the repeated index. Thus aibi means ∑
iaibi. Also, δijxj means
jδijxj = xi. When you use this convention, there is one very important thing to never forget. It is
this: Never have an index be repeated more than once. Thus aibi is all right but aiibi is not. The reason for
this is that you end up getting confused about what is meant. If you want to write ∑
iaibici it is best to
simply use the summation notation. There is a very important reduction identity connecting these two
Lemma 3.6.3 The following holds.
then every term in the sum on the left must have either
repeated index. Therefore, the left side equals zero. The right side also equals zero in this case. To see this,
note that if the two sets are not equal, then there is one of the indices in one of the sets which is not in the
other set. For example, it could be that j
is not equal to either r
Then the right side equals
Therefore, it can be assumed
then there is exactly one term
in the sum on the left and it equals 1. The right also reduces to 1 in this case. If i
exactly one term in the sum on the left which is nonzero and it must equal -1. The right side also reduces
to -1 in this case. If there is a repeated index in
then every term in the sum on the left equals zero.
The right also reduces to zero in this case because then j
and so the right side becomes
Proposition 3.6.4 Let u,v be vectors in ℝn where the Cartesian coordinates of u are
the Cartesian coordinates of v are
. Then u ⋅ v
= uivi. If u,v are vectors in ℝ3,
Also, δikak = ai.
Proof: The first claim is obvious from the definition of the dot product. The second is verified by
simply checking it works. For example,
From the above formula in the proposition,
the same thing. The cases for
are verified similarly. The last claim follows directly
from the definition. ■
With this notation, you can easily discover vector identities and simplify expressions which involve the
Example 3.6.5 Discover a formula which simplifies
From the above reduction formula,
Since this holds for all i,
it follows that