In this section we’re going to start taking a look at vectors in 2-space (normal two dimensional space) and 3-space (normal three dimensional space).  Later in this chapter we’ll be expanding the ideas here to n-space and we’ll be looking at a much more general definition of a vector in the next chapter.  However, if we start in 2-space and 3-space we’ll be able to use a geometric interpretation that may help understand some of the concepts we’re going to be looking at.

So, let’s start off with defining a vector in 2-space or 3-space.  A vector can be represented geometrically by a directed line segment that starts at a point A, called the initial point, and ends at a point B, called the terminal point.  Below is an example of a vector in 2-space.

Vectors are typically denoted with a boldface lower case letter.  For instance we could represent the vector above by v, w, a, or b, etc.  Also when we’ve explicitly given the initial and terminal points we will often represent the vector as,

where the positioning of the upper case letters is important.  The A is the initial point and so is listed first while the terminal point, B, is listed second.

As we can see in the figure of the vector shown above a vector imparts two pieces of information.  A vector will have a direction and a magnitude (the length of the directed line segment).  Two vectors with the same magnitude but different directions are different vectors and likewise two vectors with the same direction but different magnitude are different.

Vectors with the same direction and same magnitude are called equivalent and even though they may have different initial and terminal points we think of them as equal and so if v and u are two equivalent vectors we will write,

To illustrate this idea all of the vectors in the image below (all in 2-space) are equivalent since they have the same direction and magnitude.

It is often difficult to really visualize a vector without a frame of reference and so we will often introduce a coordinate system to the picture.  For example, in 2-space, suppose that v is any vector whose initial point is at the origin of the rectangular coordinate system and its terminal point is at the coordinates  as shown below.

In these cases we call the coordinates of the terminal point the components of v and write,

We can do a similar thing for vectors in 3-space.  Before we get into that however, let’s make sure that you’re familiar with all the concepts we might run across in dealing with 3-space.  Below is a point in 3-space.

Just as a point in 2-space is described by a pair  we describe a point in 3-space by a triple .  Next if we take each pair of coordinate axes and look at the plane they form we call these the coordinate planes and denote them as xy-plane, yz-plane, and xz-plane respectively.  Also note that if we take the general point and move it straight into one of the coordinate planes we get a new point where one of the coordinates is zero.  For instance in the xy-plane we have the point , etc.

Just as in 2-space, suppose that we’ve got a vector v whose initial point is the origin of the coordinate system and whose terminal point is given by  as shown below,

Just as in 2-space we call  the components of v and write,

Before proceeding any further we should briefly talk about the notation we’re using because it can be confusing sometimes.  We are using the notation  to represent both a point in 3-space and a vector in 3-space as shown in the figure above.  This is something you’ll need to get used to.  In this class  can be either a point or a vector and we’ll need to be careful and pay attention to the context of the problem, although in many problems it won’t really matter.  We’ll be able to use it as a point or a vector as we need to.  The same comment could be made for points/vectors in 2-space.

Now, let’s get back to the discussion at hand and notice that the component form of the vector is really telling how to get from the initial point of the vector to the terminal point of the vector.  For example, lets suppose that  is a vector in 2-space with initial point .  The first component of the vector, , is the amount we have to move to the right (if  is positive) or to the left (if  is negative).  The second component tells us how much to move up or down depending on the sign of .  The terminal point of v is then given by,

Likewise if  is a vector in 2-space with initial point  the terminal point is given by,

Notice as well that if the initial point is the origin then the final point will be  and we once again see that  can represent both a point and a vector.

This can all be turned around as well.  Let’s suppose that we’ve got two points in 2-space,  and .  Then the vector with initial point A and terminal point B is given by,

Note that the order of the points is important.  The components are found by subtracting the coordinates of the initial point from the coordinates of the terminal point.  If we turned this around and wanted the vector with initial point B and terminal point A we’d have,

Of course we can also do this in 3-space.  Suppose that we want the vector that has an initial point of  and a terminal point of .  This vector is given by,

Let’s see an example of this.

Okay, it’s now time to move into arithmetic of vectors.  For each operation we’ll look at both a geometric and a component interpretation.  The geometric interpretation will help with understanding just what the operation is doing and the component interpretation will help us to actually do the operation.

There are two quick topics that we first need to address in vector arithmetic.  The first is the zero vector.  The zero vector, denoted by 0, is a vector with no length.  Because the zero vector has no length it is hard to talk about its direction so by convention we say that the zero vector can have any direction that we need for it to have in a given problem.

The next quick topic to discuss is that of negative of a vector.  If v is a vector then the negative of the vector, denoted by v, is defined to be the vector with the same length as v but has the opposite direction as v as shown below.

We’ll see how to compute the negative vector in a bit.  Also note that sometimes the negative is called the additive inverse of the vector v.

Okay let’s start off the arithmetic with addition.

Below are three sketches of what we’ve got here with addition of vectors in 2-space.  In terms of components we have  and .

The sketch on the left matches the definition above.  We first sketch in v and the sketch w starting where v left off.  The resultant vector is then the sum.  In the middle we have the sketch for