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  and as we can see we get exactly the same resultant vector.  From this we can see that we will have,

The sketch on the right merges the first two sketches into one and also adds in the components for each of the vectors.  It’s a little “busy”, but you can see that the coordinates of the sum are .  Therefore, for the vectors in 2-space we can compute the sum of two vectors using the following formula.

Likewise, if we have two vectors in 3-space, say  and , then we’ll have,

Now that we’ve got addition and the negative of a vector out of the way we can do subtraction.

If we make a sketch, in 2-space, for the summation form of the difference we the following sketch.

Now, while this sketch shows us what the vector for the difference is as a summation we generally like to have a sketch that relates to the two original vectors and not one of the vectors and the negative of the other.  We can do this by recalling that any two vectors are equal if they have the same magnitude and direction.  Upon recalling this we can pick up the vector representing the difference and moving it as shown below.

Finally, if we were to go back to the original sketch add in components for the vectors we will see that in 2-space we can compute the difference as follows,

and if the vectors are in 3-space the difference is,

Note that both addition and subtraction will extend naturally to more than two vectors.

The final arithmetic operation that we want to take a look at is scalar multiples.

Here is a sketch of some scalar multiples of a vector v.

Note that we can see from this that scalar multiples are parallel.  In fact it can be shown that if v and w are two parallel vectors then there is a non-zero scalar c such that , or in other words the two vectors will be scalar multiples of each other.

It can also be shown that if v is a vector in either 2-space or 3-space then the scalar multiple can be computed as follows,

At this point we can give a formula for the negative of a vector.  Let’s examine the scalar multiple, .  This is a vector whose length is the same as v since  and is in the opposite direction of v since the scalar is negative.  Hence this vector represents the negative of v.  In 3-space this gives,

and in 2-space we’ll have,

Before we move on to an example let’s get some properties of vector arithmetic written down.

The proof of all these comes directly from the component definition of the operations and so are left to you to verify.

At this point we should probably do a couple of examples of vector arithmetic to say that we’ve done that.

There is one final topic that we need to discus in this section.  We are often interested in the length or magnitude of a vector so we’ve got a name and notation to use when we’re talking about the magnitude of a vector.

In the 2-space case the formula is fairly easy to see from a geometric perspective.  Let’s suppose that we have  and we want to find the magnitude (or length) of this vector.  Let’s consider the following sketch of the vector.

Since we know that the components of v are also the coordinates of the terminal point of the vector when its initial point is the origin (as it is here) we know then the lengths of the sides of a right triangle as shown.  Then using the Pythagorean Theorem we can find the length of the hypotenuse, but that is also the length of the vector.  A similar argument can be done on the 3-space version.

From above we know that cv is a scalar multiple of v and that its length is |c| times the length of v and so we have,

We can also get this from the definition of the norm.  Here is the 3-space case, the 2-space argument is identical.

There is one norm that we’ll be particularly interested in on occasion.  Suppose v is a vector in 2-space or 3-space.  We call v a unit vector if .

Let’s compute a couple of norms.

We now need to take a look at a couple facts about the norm of a vector.

Proof : The proof of the first part of this comes directly from the definition of the norm.  The norm is defined to be a square root and by convention the value of a square root is always greater than or equal to zero and so a norm will always be greater than or equal to zero.

Now, for the second part, recall that when we say “if and only if” in a theorem statement we’re saying that this is kind of a two way street.  This statement is saying that if  then we must also have  and in the reverse it’s also saying that if  then we must also have .  To prove this we need to make each assumption and then prove that this will imply the other portion of the statement.

We’re only going to show the proof for the case where v is in 2-space.  The proof for in 3-space is identical.  So, assume that  and let’s start the proof by assuming that .  Plugging into the formula for the norm gives,

As shown, the only way we’ll get zero out of a square root is if the quantity under the radical is zero.  Now at this point we’ve got a sum of squares equaling zero.  The only way this will happen is if the individual terms are zero.  So, this means that,

So, if  we must have .

Next, let’s assume that .  In this case simply plug the components into the formula for the norm and a quick computation will show that  and so we’re done.

Proof : This is a really simple proof, just notice that u is a scalar multiple of v and take the norm of u.

Now we know that  because norms are always greater than or equal to zero, but will only be zero if we have the zero vector.  In this case we’ve explicitly assumed that we don’t have the zero vector and so we now the norm will be strictly positive and this will allow us to drop the absolute value bars on the norm when we do the computation. 

We can now do the following,

So, u is a unit vector.

This theorem tells us that we can always turn a non-zero vector into a unit vector simply by dividing by the norm.  Note as well that because all we’re doing to compute this new unit vector is scalar multiplication by a positive number this new unit vector will point in the same direction as the original vector.