To begin with consider u ×

Next consider u×

When you multiply this out, you get


and if you are clever, you see right away that

Thus
 (3.32) 
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 and 3.33 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 follows.There are two special symbols, δ_{ij} and ε_{ijk} 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 δ_{ij}, called the Kroneker delta symbol is defined as follows.

With the Kroneker symbol i and j can equal any integer in
Definition 3.6.2 For i,j, and k integers in the set,

The subscripts ijk and ij in the above are called indices. A single one is called an index. This symbol ε_{ijk} is also called the permutation symbol.
The way to think of ε_{ijk} is that ε_{123} = 1 and if you switch any two of the numbers in the list i,j,k, it changes the sign. Thus ε_{ijk} = −ε_{jik} and ε_{ijk} = −ε_{kji} 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 ε_{iij} = −ε_{iij} and so ε_{iij} = 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 a_{i}b_{i} means ∑ _{i}a_{i}b_{i}. Also, δ_{ij}x_{j} means ∑ _{j}δ_{ij}x_{j} = x_{i}. 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 a_{i}b_{i} is all right but a_{ii}b_{i} is not. The reason for this is that you end up getting confused about what is meant. If you want to write ∑ _{i}a_{i}b_{i}c_{i} it is best to simply use the summation notation. There is a very important reduction identity connecting these two symbols.
Proof: If
Therefore, it can be assumed
Proposition 3.6.4 Let u,v be vectors in ℝ^{n} where the Cartesian coordinates of u are

Also, δ_{ik}a_{k} = a_{i}.
Proof: The first claim is obvious from the definition of the dot product. The second is verified by simply checking it works. For example,

and so

From the above formula in the proposition,

the same thing. The cases for
With this notation, you can easily discover vector identities and simplify expressions which involve the cross product.
Example 3.6.5 Discover a formula which simplifies
From the above reduction formula,
