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Let’s start this section off with a quick discussion on what
vectors are used for. Vectors are used
to represent quantities that have both a magnitude and a direction. Good examples of quantities that can be
represented by vectors are force and velocity.
Both of these have a direction and a magnitude.
Let’s consider force for a second. A force of say 5 Newtons that is applied in a particular
direction can be applied at any point in space.
In other words, the point where we apply the force does not change the
force itself. Forces are independent of
the point of application. To define a
force all we need to know is the magnitude of the force and the direction that
the force is applied in.
The same idea holds more generally with vectors. Vectors only impart magnitude and
direction. They don’t impart any
information about where the quantity is applied. This is an important idea to always remember
in the study of vectors.
In a graphical sense vectors are represented by directed
line segments. The length of the line
segment is the magnitude of the vector and the direction of the line segment is
the direction of the vector. However,
because vectors don’t impart any information about where the quantity is applied
any directed line segment with the same length and direction will represent the
same vector.
Consider the sketch below.
Each of the directed line segments in the sketch represents
the same vector. In each case the vector
starts at a specific point then moves 2 units to the left and 5 units up. The notation that we’ll use for this vector
is,
and each of the directed line segments in the sketch are
called representations of the
vector.
Be careful to distinguish vector notation, 
,
from the notation we use to represent coordinates of points, . The vector denotes a magnitude and a
direction of a quantity while the point denotes a location in space. So don’t mix the notations up!
A representation of the vector in two dimensional space is any directed line
segment, ,
from the point to the point . Likewise a representation of the vector in three dimensional space is any directed
line segment, ,
from the point to the point .
Note that there is very little difference between the two
dimensional and three dimensional formulas above. To get from the three dimensional formula to
the two dimensional formula all we did is take out the third
component/coordinate. Because of this
most of the formulas here are given only in their three dimensional
version. If we need them in their two
dimensional form we can easily modify the three dimensional form.
There is one representation of a vector that is special in
some way. The representation of the
vector that starts at the point and ends at the point is called the position vector of the point . So, when we talk about position vectors we
are specifying the initial and final point of the vector.
Position vectors are useful if we ever need to represent a
point as a vector. As we’ll see there
are times in which we definitely are going to want to represent points as
vectors. In fact, we’re going to run
into topics that can only be done if we represent points as vectors.
Next we need to discuss briefly how to generate a vector
given the initial and final points of the representation. Given the two points and the vector with the representation is,
Note that we have to be very careful with direction
here. The vector above is the vector
that starts at A and ends at B.
The vector that starts at B
and ends at A, i.e. with representation is,
These two vectors are different and so we do need to always
pay attention to what point is the starting point and what point is the ending
point. When determining the vector between two points we always subtract the
initial point from the terminal point.
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Example 1 Give
the vector for each of the following.
(a) The
vector from to .
(b) The
vector from to .
(c) The
position vector for
Solution
(a) Remember
that to construct this vector we subtract coordinates of the starting point
from the ending point.
(b) Same thing
here.
Notice that the only difference between the first two is
the signs are all opposite. This
difference is important as it is this difference that tells us that the two
vectors point in opposite directions.
(c) Not much to
this one other than acknowledging that the position vector of a point is
nothing more than a vector with the points coordinates as its components.
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We now need to start discussing some of the basic concepts
that we will run into on occasion.
Magnitude
We also have the following fact about the magnitude.
This should make sense.
Because we square all the components the only way we can get zero out of
the formula was for the components to be zero in the first place.
Unit Vector
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Any vector with magnitude of 1, i.e. ,
is called a unit vector.
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Example 3 Which
of the vectors from Example 2 are unit vectors?
Solution
Both the second and fourth vectors had a length of 1 and
so they are the only unit vectors from the first example.
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Zero Vector
The vector that we saw in the first example is called a
zero vector since its components are all zero.
Zero vectors are often denoted by . Be careful to distinguish 0 (the number) from
(the vector).
The number 0 denote the origin in space, while the vector denotes a vector that has no magnitude or
direction.
Standard Basis
Vectors
The fourth vector from the second example, ,
is called a standard basis vector. In three dimensional space there are three
standard basis vectors,
In two dimensional space there are two standard basis
vectors,
Note that standard basis vectors are also unit vectors.
Warning
We are pretty much done with this section however, before
proceeding to the next section we should point out that vectors are not
restricted to two dimensional or three dimensional space. Vectors can exist in general n-dimensional
space. The general notation for a
n-dimensional vector is,
and each of the ai’s
are called components of the
vector.