In this section we need to have a brief discussion of vector
We’ll start with addition
of two vectors. So, given the vectors and the addition of the two vectors is given by
the following formula.
The following figure gives the geometric interpretation of
the addition of two vectors.
This is sometimes called the parallelogram law or triangle
is very similar. Given the vectors and the difference of the two vectors is given by,
Here is the geometric interpretation of the difference of two
It is a little harder to see this geometric
interpretation. To help see this let’s
instead think of subtraction as the addition of and . First, as we’ll see in a bit is the same vector as with opposite signs on all the
components. In other words, goes in the opposite direction as . Here is the vector set up for .
As we can see from this figure we can move the vector
representing to the position we’ve got in the first figure
showing the difference of the two vectors.
Note that we can’t add or subtract two vectors unless they have the same number of components. If
they don’t have the same number of components then addition and subtraction
can’t be done.
The next arithmetic operation that we want to look at is scalar multiplication. Given the vector and any number c the scalar multiplication is,
So, we multiply all the components by the constant c.
To see the geometric interpretation of scalar multiplication let’s take
a look at an example.
Example 1 For
the vector compute ,
and . Graph all four vectors on the same axis
Here are the three scalar multiplications.
Here is the graph for each of these vectors.
In the previous example we can see that if c is positive all scalar multiplication
will do is stretch (if ) or shrink (if ) the original vector, but it won’t
change the direction. Likewise, if c is negative scalar multiplication will
switch the direction so that the vector will point in exactly the opposite
direction and it will again stretch or shrink the magnitude of the vector
depending upon the size of c.
There are several nice applications of scalar multiplication
that we should now take a look at.
The first is parallel vectors. This is a concept that we will see quite a
bit over the next couple of sections.
Two vectors are parallel if they have the same direction or are in
exactly opposite directions. Now, recall
again the geometric interpretation of scalar multiplication. When we performed scalar multiplication we
generated new vectors that were parallel to the original vectors (and each
other for that matter).
So, let’s suppose that and are parallel vectors. If they are parallel then there must be a
number c so that,
So, two vectors are parallel if one is a scalar multiple of
The next application is best seen in an example.
Example 3 Find
a unit vector that points in the same direction as .
Okay, what we’re asking for is a new parallel vector
(points in the same direction) that happens to be a unit vector. We can do this with a scalar multiplication
since all scalar multiplication does is change the length of the original
vector (along with possibly flipping the direction to the opposite
Here’s what we’ll do.
First let’s determine the magnitude of .
Now, let’s form the following new vector,
The claim is that this is a unit vector. That’s easy enough to check
This vector also points in the same direction as since it is only a scalar multiple of and we used a positive multiple.
So, in general, given a vector ,
will be a unit vector that points in the same
direction as .
In the previous section we introduced the idea of standard
basis vectors without really discussing why they were important. We can now do that. Let’s start with the vector
We can use the addition of vectors to break this up as
Using scalar multiplication we can further rewrite the
Finally, notice that these three new vectors are simply the
three standard basis vectors for three dimensional space.
So, we can take any vector and write it in terms of the
standard basis vectors. From this point
on we will use the two notations interchangeably so make sure that you can deal
with both notations.
We will leave this section with some basic properties of
The proofs of these are pretty much just “computation”
proofs so we’ll prove one of them and leave the others to you to prove.
with the two vectors, and and yes we did mean for these to each have n components. The theorem works for general vectors so we
may as well do the proof for general vectors.
Now, as noted
above this is pretty much just a “computational” proof. What that means is that we’ll compute the
left side and then do some basic arithmetic on the result to show that we can
make the left side look like the right side.
Here is the work.