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A comprehensive guide to vectors in the plane and 3d space, covering fundamental concepts, operations, and applications. It explains vector addition, subtraction, scalar multiplication, dot product, cross product, and their geometric interpretations. The document also explores lines and planes in 3d space, including parametric and symmetric equations, distance calculations, and cylindrical surfaces. It includes numerous examples and illustrations to enhance understanding.
Typology: Lecture notes
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Sections11.
Vectors in the
Plane
vector in a plane
is a
directed
line segment
betweentwo
points
points
pi
Pa
q
vector
is the
vector
staring
and ending
atQ
p
pa
vector has
direction
and
length
Q
5 Q
1,
E Iiiii
quivalent
Vectors
Two
vectors
are equivalentif they
have
the
same
direction and
length
info 1
direction
2
length
omponentform
component
form
vector is
y
coordinates
when
placed
with the starting
point
of the
origin
PQ
31
3 3 isthe
component
form
of PQ
Length
qfayu.fi
EIfV
v
Va
in
componetform
then
the length
denoted
by
In
eiyf.IE i
isii
1
Addition
Vatwa
at
ouromens
e
s
Subtraction
Vw̅ v
W
Va
Wa
E
w̅
direction
Scalar
Multiplication
R
and J v
Ja
be a vector
CJ CV C
V
changes direction
if
c is negative
V 1 1
if c o
25
2,
2gÉejft E
E
for the real
numbers
Addition
is
commutative
to
Associative
Scalaris
commutative
and
associative
a b c ER
ab
ba i
b eV a
b
Tectoffs
silly
a ve
Ing iÑÉ
ual
E
D
vector
till
roof l cill
Eternal
Futura
Ic
Fta
c
11011
Dimen
sionalVectg Standard
Vectors
3D
Vectors
in
still
determined
by
V
length
and
direction
ull
Tv
lengthfor
ength
for
Vectors
mail.FI
E
ii
ts
Find the length
of the
vectorPT
where
p
a
14
go.ee
erforiginIIPQT
MH35 MaH
T
ector Algebra
Remains the same in all
dimensions
rite v
VaV
4 41,
43
Utu u.tv
U2V2Ust
V
thenCv Cv
Cva
cuz
vector
1
U
3,
Citi 6,
f
0
to
1 6 6 1 2 3
tw
u
utw̅
x P 5
0,
3 PQ
5,
Find a
sphere
where
the
center
11
5 3636 V
the
midpoint
of PT
and with
line
segment
being
a
diameter
r
v
Midpoint
5
2
1,
r.mg
9cf
aEPzhs
between
points
Xo
yo
20
and x
z
distance
EHg
y.DE
ygntgedatCxo
zo
radius r
of
points
of constant distance
equation
for
centered
Xo
yo
20
of
r is
x
xo
2
20 r
between
and
Midpoint
is
P
PQ
7 p pj
Pi
Pi
92
Ps
PII
PII
PII
f
If
here
s
career's.tn
thatbeup.IE
x Let w have
intial
2
terminal
point
is
4j
K
parallel
to w
Find w in component
form
1
3 6
the
Eat
taw
then
6,
130
355
its
are
parallel
Project u
onto
v
a
projule
Hullcoso
1
II
Y
cos Iprojuall
11411
example
Forcedistan
W
F
d
Puffit
EffHow
much work
did
the
farmer
do
the door
pros
T
F IF
1
50
cost
sin 601
E
F.EC
E'stt
i
25
bsft
Cross
Product
Let 4
41,42 43
v u
Va
V
product
of u and v
to be
a
tl
5
it
I
11
u
xv
a
v
v
u
Jj
a
v
viualk
Ex 4
u u
talk
Troperties
of Cross
Product
axv vxu
2 ax vta uxutuxw
casva.cat
www.t
fittj
L
itam
uxv is perpendicular
to u and v with
0
respect
to
the
hand
rule
ataxv
a xu 0
nuxu
E.TT
remember
i 1 0,
J
Corio
creadia
Geometry of
Cross
Product
lemmata
d
it is
angle
between u v
HullHull
sin Hall
2110112
f
Hall
Hull
5
triffid 14
2
12
caravans cars
Lx
Suppose
that
a force of 50lb is imparted
to a lever
that
makes
angle
with the x axis
How muchtorque is
there if the force is
directed
straight
down
able
control
15
État.si
oj
5Bj
so
151
Product
et u v
w be
three vectors
triple
product is
u
v w a uxw
v wxu w
uxv
Y Y
0243
4203
42
v w
Uzw Uz
u
wa
w
v
Thm The
product
gives
absolute
vale of
the
parallelepiped
was.ie
iiijiii
volume
Area of
like
angle
between w
and uxv
X
Find
the
volume of
the
parallelepiped
u
1
v
volume
u
uxw
1
8
0
0
3
Lines
Planes
on to
describe
in
space
Point
on
pi Pa
Ps
Direction vector
v v
parametric
form
pi
pitty
Patva
P
Pictor
pathe
line
I
IE
Symmetric
Equations
for a line
FFTTeline
through
0,
Cartesian
solve for
t and
get
equations in
direction v
1,
involving x
2
YEI.EE
nsnespt
Y
YPa
2
9
170
with
t o
6
Example
Take
p
0,
2,
IF If
v
x
p
v.ly
find
two points
qq.gg
y
pa
v
z
p
PT
Each
of these
describes
a plane
Y
Therefore
this
gives
theequation for
a line as
the
intersection of two
planes
Pi Pa Ps
is a point
plane
PQ
ñ
is equivalent
PQ.in 0
Ty
IYgtey.lt
pI
Leto
p
sk
n
cEf
naly
paltns
2
ps
is
the
normal
origin
vector
to
plane
The white
equations
for a plane
9
on
plane
n x
thaly
pa
this
z
ps
a
point
P in the
plane
Parametric
form
vectors
u v
insidethe plane
2 the
normal vector ñ
which are not parallel to
each other
Let
p
be
a
point
in the plane
Pitatsu
t
s
x̅ x
z
parameters
eatin
is
aao
Cylindrical
Surfaces
Ruled Surfaces
These are
surfaces
made
involve
only
two
of the
Cartesian
coordinates and
coordinates
is
missing
Ex x
ty
In
the x
is
a
circle of a radius
at
the
origin
Notice that in
2 2
gives
the
circle
m
y
f
x
2
92
y
1
Y
1
Ellipsoid
He
I
x
221
2ty
V
One sheet
hyper
beliad
In To
at the
cur
y
2
22
92
22 21
FEY the
ya
plane
the
tin
radius is K
e
circles
