What is a vector and how do I calculate the 'modulus' of a vector?

A vector is something that has both magnitude and direction. An example of a vector is velocity because it has magnitude and direction. However, speed is not a vector. This is because it only has magnitude. A vector is represented by OA (with an arrow above it going from the O to the A) or a. In 3-D there are 3 main unit vectors: i, j, k. Where:
i = (1, 0, 0) , j = (0, 1, 0) and k = (0, 0, 1). All 3-D vectors are represented in terms of these three unit vectors. (Unit vectors all have a length of 1). So we can say the general form of a vector, for some vector called a, is: a = xi + yj + zk = (x , y, z) where x, y and z are numbers. Let us try to visualise vector a: 
In order to find vector a we must move x units in the x (or i) direction, y units in the y (or j) direction and then z units in the z (or k) direction. The magnitude of a vector is known as its modulus (or more simply, length). The modulus of vector a = ( x² + y² + z²)^1/2