This

**preview**shows half of the first page. to view the full**3 pages of the document.**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 ~

QR.

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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