Notation and the components of a vector

Before we can discuss the analytical approach to dealing with vectors, we need to learn how to write them down. One method has already been mentioned. This uses the relationship between a position vector and a point in to write a vector as an ordered n-tuple. For instance, the vector (3,-1,4) points three units toward the x axis, one unit down the y axis, and 4 units along the positive z axis. As we move into higher dimensions than three, we just add more numbers to the list. So, in , the vector is just the same as the ordered n-tuple where the numbers are called the components of the vector .

Another method of writing vectors involves the ideas of scalar multiplication and the addition of vectors. We start by creating a set of vectors called basis vectors. These vectors have two important properties.

1. The basis vectors are all unit vectors. This means that they each have a length (or magnitude) of exactly 1. Usually we write unit vectors with a ``hat'' over them instead of an arrow. So if is a unit vector, we would usually write it as .
2. There is one basis vector for each direction in which we can move. In three dimensions, this gives us three basis vectors, one along each coordinate axis. The basis vector that points in the direction of the +x axis is called . The vector pointing toward the +y axis is called and the vector in the z direction is called .

   

If we want a vector that points three units along the +x axis, we can simply multiply the vector by the scalar 3 to get . We can do similar things to get the vectors in the y and z directions. Now, if we add up these vectors, we'll have a vector which points out in some direction determined by the components of the vector. Thus, the vector (3, -1, 4) could be written as .