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MATH104 Chapter Notes -Parallelogram Law

Course Code
Steve Spencer

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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:
Definition The directed line segment from a point Pin R2to a point Qin
R2is drawn as an arrow with starting point Pand tip Q. It is denoted by ~
P Q.
Example: Let P= (4,4), Q= (3,6), R= (6,2), and O= (0,0). Then
the following figure illustrates the directed line segments ~
OP ,~
P Q, and ~
Now, the directed line segment ~
OP is the same as our visualization of the
vector ~p =4
4. (Sidenote: For some reason, “points” get capital letters,
while “vectors” get lowercase letters. It will happen frequently that the point
“P” will suddenly become the vector “~p”. We will often switch back and forth
between considering an element of R2to 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 R2.)
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