If we label the axes as 1,2, and 3 we can write the dot product as a sum
If we number the elements of a matrix similarly,
we can write similar expressions for matrix multiplications
Notice that in each case we are summing over the repeated index. Since this is so common, it is now conventional to omit the summation sign.
Instead we simply write
We can then also number the unit vectors, êi, and write
which can be convenient in a rotating coordinate system.
The Kronecker delta is
This is the standard way of writing the identity matrix.
Levi-Civita (Alternating) symbol
Another useful quantity can be defined by
With this definition it turns out that
This will let us write many formulae more compactly.