Vectors in the Plane
John E. Gilbert, Heather Van Ligten, and Benni Goetz
Calculus for functions
z
=
f
(
x, y
)
of two (or more) variables relies heavily on what you already know
about the calculus of functions
y
=
f
(
y
)
of one variable. A few preliminary ideas about vectors, various
coordinate systems in two and three variables, as well as morecomp
l
icatedcurvesinthep
laneones
usually deFned implicitly  need to be studied too. It’s convenient to do this in terms of
vector
functions
.Bu
ton
c
ew
e
’v
edon
etha
t
,ca
l
cu
lu
si
sth
en
ex
ts
t
ep
!
Let’s start with vectors  you may have met them already, and you’ll certainly make good use of
them in a number of your other courses!
What is a vector:
Aquant
ity
,beitgeometr
ic
,sc
ient
iFco
rwhatever
,isa
vector
so long as it has both a
magnitude
(or
length
)anda
direction
.±o
rin
s
tance
,
velocity
can be described by a vector because it has a
magnitude, namely
speed
,aswe
l
lasad
irect
ion
:thew
indb
lowsataspeedo
f5mphfromthe
northwest, Joe heads due north at 75 mph in his car, and so on.
Displacements
provide a di²erent type of
example: let’s look at where Bob lives in relation to
Alice. His house is at point
B
which is
223
ft.,
18
◦
ENE, from Alice’s house at point
A
.I
fw
erep
re
s
t
this as an arrow from
A
to
B
,itdete
rm
inesa
displacement vector
→
AB
with
magnitude
the
distance from
A
to
B
,and
direction
the direction
from
A
to
B
t
’
sna
tu
ra
ltor
r
e
s
tth
i
sv
e
c
to
r
by an arrow with
A
the
tail
and
B
the
head
.
A
B
N
E
100
ft
In general, we’ll usually label vectors by single boldfacedlettersl
ike
a
,
v
,.
.
.,andsoon
.Bewa
:
physicists and engineers sometimes use di²erent notation. The
length
of a vector
v
will be denoted by

v

;th
islengthisapos
it
ivenumberexceptfo
rthe
zero
vector
0
which has length
0
.O