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Unformatted text preview: 24 CHAPTER 1. COORDINATES AND VECTORS →
−
w →
−
w
→+
−
v
→
−
v
+→
−
w →
−
v
→
−
w →
−
v
Figure 1.19: Parallelogram Rule (Commutativity of Vector Sums) ﬁnally, suppose mi
→ℓ
ni →
is a convergent sequence of rationals. For any ﬁxed vector − , if we draw
v
→
− with all their tails at a ﬁxed
arrows representing the vectors (mi /ni ) v
position, then the heads will form a convergent sequence of points along a
→
line, whose limit is the position for the head of ℓ− . Alternatively, if we
v
→
− and any positive real number
pick a unit of length, then for any vector v
→
→
→
r , the vector r − has the same direction as − , and its length is that of −
v
v
v
multiplied by r . For this reason, we refer to real numbers (in a vector
context) as scalars.
If
→→→
− =− +−
u
v
w
then it is natural to write →→→
− =− −−
v
u
w →
w
and from this (Figure 1.20) it is natural to deﬁne the negative −− of a
→
− as the vector obtained by interchanging the head and tail of − .
→
vector w
w
→
− by any negative
This allows us to also deﬁne multiplication of a vector v
real number r = − r  as
→
→
(− r )− := r  (−− )
v
v
→
—that is, we reverse the direction of − and “scale” by r .
v
Addition of vectors (and of scalars) and multiplication of vectors by scalars
have many formal similarities with addition and multiplication of numbers.
We list the major ones (the ﬁrst of which has already been noted above):
• Addition of vectors is →→→→
commutative: − + − = − + − , and
v
w
w
v ...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
 Fall '08
 ALL
 Calculus, Vectors

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