What is Orthonormal?

Two vectors x and y are orgonal and each vector has the length of one are said to be orthonormal.
length of x = sqrt(<x, x>) = 1
or simply <x, x> = 1
<y, y> = 1
<x, y> = 0

What is Orthogonal?

Two vectors x and y are described as orthogonal if their inner product is zero:
<x, y> = 0

Inner Product

The inner product is denoted by <x, y> where x and y are vectors.
If x = (1, 2, 3) and y = (6, 5, 4), then
<x, y> = (1)(6) + (2)(5) + (3)(4) = 28.
It is obvious that <x, y> is a scalar.

A Field

A field (F, +, .) is a set F with operations `+` (or addition) and `.` (or multiplication) satisfying the properties including closure for addition (x + y also in F),
commutativity for addition (x + y = y + x),
associativity for addition (x + (y+z) = (x+y) + z),
existence of additive identity (x + 0 = x),
existence of additive inverse (-x),
closure for multiplication (x.y also in F),
commutativity for multiplication (x.y = y.x),
associativity for multiplication (x.(y.z) = (x.y).z),
existence of multilicative identity (x.1 = x),
existence of multiplicative inverse (1/x), and
distributive law (x.(y+z) = (x.y) + (x.z)).

The set of Integers

The set of integers is considered as the set of natural numbers, zero, the negative of natural numbers. This is {..., -3, -2, -1, 0, 1, 2, 3, ...}.
It is denoted by Z.

Natural Numbers

The natural numbers {1, 2, 3, 4, ...} usually denotes by N.

The Euclidean norm (or length)

A convenient measure of the "length" of a vector x is the Euclidean norm:
|x| = norm of x = (x,x)^(1/2)

The product of two n-dimensional vectors

(x,y) = x1 y1 + x2 y2 + ... + xn yn = x y

Share DVD drive under OS X environment

To share the DVD drive in your Mac with Windows XP network, you need to create a symbolic link.

First, open a Terminal window in you OS X. Second, execute the following command:
$ ln -s /Volumes symlinkDir
where "symlinkDir" is the symbolic link directory that you are going to use.

Finally, your Windows XP could to go your shared directory. Through the "symlinkDir", you can access the DVD drive in your Mac.