Here is a proof of the distributive law for the cross product. Let x be a vector. From the above observation,
x ⋅ a×  =
⋅ =
⋅ b+
⋅ c 
= x ⋅ a × b + x ⋅ a × c = x⋅
. 
Therefore,

for all x. In particular, this holds for x = a×
First, here is some useful notation. 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 12.1.7 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 12.1.8 For i,j, and k integers in the set,

The subscripts i,j,k 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
Definition 12.1.10 Let u ≡

where the determinant is expanded formally along the top row. Let f : U → ℝ^{3} for U ⊆ ℝ^{3} denote a vector field. The curl of the vector field yields another vector field and it is defined as follows.

where here ∂_{j} means the partial derivative with respect to x_{j} and the subscript of i in

Note the similarity with the cross product. More precisely and less evocatively,

In the above, i = e_{1},j = e_{2}, and k = e_{3} the standard unit basis vectors for ℝ^{3}. Using the nice notation, ∇×

where ∂_{j} denotes the differential operator which comes by taking the derivative with respect to x_{j}.
The following gives the description of the cross product using this nice notation.
Proposition 12.1.11 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 that it works. For example,

and so

From the above formula in the proposition,

the same thing. The cases for