Lecture 1c
Directed Line Segments
(pages 7-8 )
So far, we have always had our vectors \start" at the origin, and \end" at
the point corresponding to our vector. But if we are thinking of vectors as a
direction instead of as a point, then it shouldn’t really matter where we start or
end. And so, we will expand our study of vectors to now consider the following:
De▯nition The directed line segment from a point P in R to a point Q in
2
R is drawn as an arrow with starting point P and tip Q. It is denoted by PQ.
Example: Let P = (4;4), Q = (▯3;6), R = (▯6;▯2), and O = (0;0). Then
the following ▯gure illustrates the directed line segments OP, PQ, and QR.
Now, the directed line segment OP is the same as our visualization of the
▯ ▯
vector ~ = 4 . (Sidenote: For some reason, \points" get capital letters,
4
while \vectors" get lowercase letters. It will happen frequently that the point
\P" will suddenly become the vector2\p~". We will often switch back and forth
between considering an element of R to be a point or a vector, and perhaps this
change in notational convention helps keep straight which particular aspects we
are trying to emphasize or make use of. Nevertheless, the point P, vector p~,
and directed line segment OP are all the same element of R .)
1 De▯nition A directed line segment that starts at the origin and ends at a point
P is called the position vector for P.
And so, you may be asking, what do directed line segments have to do with our
study of vectors? Sure, those position vectors gave us a nice parallelogram rule
for addition, and putting vectors end-to-end when adding was fun too. But if
vectors are the same as points in R , then what is PQ?
Well, part of the idea behind viewing points as vectors is to realize that to
2
describe a point in R , you need two pieces of information: its distance from the
origin, and which direction to travel that distance. So the key bits of information
about a vector are its length and direction. But directed line segments do not
increase our collection of lenghts/directions{they simply move vectors around
so that they start at a point other than the origin. And so, we shall consider
two directed line segments to be (more or less) the same if they have the same
length and direction. That is...
~ ~
De▯nition We de▯ne two directed line segments PQ and RS to

More
Less