Class notes Calculus III

Introduction
to Vectors
-Vectors have both a speed and a direction . * Vector i W/ initial point at the
origin ,
and

↳ "Speed" Magnitude of the terminal at P(v, va) called
given by length
: the rector
. a point is a
,




position rector of PCV , Va) and is shown [V , Va]
-Vectors system
are
mapped on a coordinate
*
Any vector can be translated into a
position vector
.




Example
to do this :

B Vector : v = AB Suppose two points are
given
4 3X
,
,
%) is PlXa 2) ,




O




(xz
A
,
is ")
>
u = PP = - x
,,
+2 +
8
slope :
Ye
2
rectori : = [D (12) ( 2) Length /magnitude) :
> .




↳ IIv/l 1 V =
D I

theorem
I 1 1 1 1 I I
(pythagorean
I




Can multiply by a constant :
Scalars v: Slope-5 Kill ,
=
2s =
-



Change the length of the rector (magnitude) W :
Slop = Kill : S : 25 = 29
-
Reverse the "direction" of the vector
. v =
[




i 38

A faster Find the vector of both
way position
:
⑨
.




⑤

#king with position
rectors



-v =, a
Y
·
-
T
⑤ ,




+ 5 =
19 06 ,
,,
9.36 / ,

-


Scalar multiples are Parallel
For scalar
,
C V . =
XC . V
,
C .


Vay
Adding subtracting
-and rectors

Examplea =
< 1
,
2) ,
5 13 1) =

,




↑ o N &

W
* vo = 00
&



2 2) 72 4)
= (2 - -
1
,
2 . =
,



a =5 1 103 201) 12 37
Y
= - =

, ,



* a b 1 1 3 2 1) 1 4 1)
- =
-
-



,
- = -



,

& &
1125 51 (2 103 2 201) <1 5) 526
+ = . -



,
.
=
,
=




(parallelagrum law)
* -
To

*
& *
* v - w = vb) w)-

,2




Un
Vectors
A rector with a length of
↳ Divide
position vector by it's a 34645 5 45 2j Find & where llill
magnitude 3
=
a =
·
=
,

i
↳ Unit vector : =

Full and //2 -
35



Sooo ...
= Kill ·, therefore ,
i denotes the direction .
2 - 35 = c = (2 . -
3 -
3 1 .




,
2 4 .
-

3 .

2) =
79 27 ,



i = -



94625

1) 11011 Strjoi 2 90 2)
-




i = (3 + = = 2 =
,




u = 1) = < b) ,



v =
3 -
94 2j)
v [ ,t)
i = - = -
magnitude want change simply
,



distribute the
negative.



= cosET o
SinGy
in

&
Mandard Basis Vectors Find i such that I ill = 9




i =
X 1 8) (x direction) in
T


78
i
T
and makes
Th

- with the
an


x-axis
angle of


↑
.

,

cost
3
I




j
=
<0 17 (y direction)
,


u =
cost sinTy = 524022
v =
<V ,, k) =
[v 0700 v = , ,
u = 9(50zj)
=
<1 07 o va/O 1
v
, , ,


Y v
TbV]
=




i
, Y
,




Example v = <3 ,2)
F E n = -
(E ,+ E))

2LB
- I
i = 34 -



25 w



· = (3 ,
-
2)
,
5 (2 =
,
6) ,
i = 4 1) ,
~ = cost + sin (0 = =
j) .
2 =
2y
F - 1E 1) (cost o sinj) 1 11) Ei
=

,
= -



,
-



Ei)
SBV : =
34 -


2y ,
b =
9i -

6 ,
i =
35 F IE1l(cos =sin j) 1 (l) Ei
= =
-
-


i)
Slope : ma =
-E ,
mi
=
-Ez me (parallel rectors)
I
=




Mayn : Hall = 53
,
11511 = , Hell +
2j = + [ -
11E , 1)( 2 Si) 11 =(1) Si ti))
-
-
-




a + 5 + E 2 =
I
* If vectors shee the same i they will be

Wo = E llllll
,
-




parallel . ,



↳ All rectors sealed multiples
are
just a



(scaler multiples) of a certain unit reator. = 2I ,



5 = 3 i =

lol=S
,




11 11 ,
:
E
4llE211 : 4 -
IIE11 / =

,3



(11)
Using
Vectors #ample 2x2 + 2+ = = 2z' bx -

4y - 22 -
1 =
0


i = 500(cos45
°
+ + Sin45j) 2x 6x +242 472z32z = 1

↳
soomph v = 80(cos15o >
sin15oy) x -
3x -y +
y 2y + / +
zzt = 2 +
-31t
somph (x z) (y 13(z z)
-
+ - + = 4
W

~
453
radius : r = 2

725 center : (2 1 2)
,
,




Find
Example eg .
for sphere where Al2 3 4) , ,
,
B13 2 1) , ,




at opposite ends of diameter

Vectors
are a


3-D in Center will be the midpoint.
center : (E :E , )
z(x , y ,
z) ; (2 ,
3
,
4) radius : Ed(AB) S :




E) v(y z)(z E)
X

3
-




= (x -
+ -




"
·
-




Vectors
&


1-3
-




, in



*
-




Y
-




-
4
Ration
Vectors

(V ., Va v)
-




↑= V + Vai
+X ,
or =
,
, vay ,




Z
HillNussus
(noHal)
-




* Parallel rectors are
always scalar multiples .




all'd iff b =
c . a

Distance : 4
,
(X ,, % ,
z
,
) 3 .
P(Xa ya 2) , ,




Example Show : = i -



2jo5k is 11 &
d)p p.) J(x,
,
= -
x
,
) -(+ 2
-

y, )s(z - z
,
) 5 =
(3 ,
-

6
,
15) =
371 ,
-2
,
5)
Show that A (3 4 , ,
1) ,
B (4 ,
4, 6) (13
,
,
1
,
2) Parallel
form an isosceles triangle
5 =
-Y -



Ej -k =
j(i ,
2jsSi)
d(AB) 05 = = 526
Parallel
d(BC) = S = 526
d(AC) 553
:

= % So =
Example i = T -

2j ,
5 =
(2 ,
3
,
17
1. 2 -
35 =
74 -



5y -
31

Adprint :
(i 2
. 113 ll =
545 =
35

3 11-2511
. =
556 =
254

CelesSpheres (X-h) o (y 1) = (z 2)
-
>
:
-
-

= ri

Example * (2 1
, ,
6) 3 B(1 ,
4
, 5) ...
find position rector B

T = -
To
3joSE


Example v = -
-
3j - 1
,
Find i

i -
1)

, "Do
Product
110-wIl =
1 w/lollwll"- 2llwll Kill · ·

coso
↳ Adds the products of
corresponding components
of two rectors and Escalar (v w) (v w). =


"gives"
-
-


a
,




a = (a ,,
22
,
an)5 [b =

,
be ,
beY ↑N lol all t
allvIIII all -

2 %. = -

cas
· ·5 = (a )(b ) o (22) (bz)
, ,
-(a)(ba) =
c v .
w =
11 will will coso
cost
all
-




Example V = 2i -

3jdk = + +
2j - 24
cost (i)
,


& =




V. = -
26 -
60 -
2 = -
10




Note Thisworks for A
Properti-
e wou ↳ If O ,
they are
parallel
perpendicular/normal
2
. vn + ) = V . +v : w
/orthogonal
.
3 ((v) ·
v =
c(v v) - = v(c a) .




↳ If 8= they scalar
i
,
are
negative
4 .
8 . v = 0 multiples
.
5 Vor =
10/1 :
vioVou?
↳ Kell = Fr v . = Kv11 ·



Kill co
·




V W O
paallels
: = this is how to are
orthapurl.
Example ,




- = X1 ,
-
3
,
2) 0 = Y 2 4
, ,
17 ,
i = 24 -



4ybi ↑, j i
>
-

mutually ordhaguel
, ,
1


4 i k y k
0 y
= .

=
.

= -




1 % (w 2) V (0 0 2) T i R.k 1
y y
. + = .
=
= =
.
, , . -




= 06834
4
=

Example
, 0? V =
2y +
3 w X, 1 = 2)
. (v w)n 124 (0)(1) (2)(1) (3)(2) c
,
w
245048y
-

2 . = 12i = -



Cost = -
= .




Goog .

5 Toto4= Mill Iall .




.
3 Il % -All olsoll =
59-11 = 70 cost =
-J ,
cos) cost ,
63 10.




Law 1, Ei Ej k
of Cosines
Example a =
2+ -


j > 3 b = -
=




a = bo i - 2bc cosA .
II or
I?
B


yb()
C

3(34
i
a wo = i a = 2 -


j - 3 =
-
=
35
Y 3
A
? = - w a = 3 .
5 = allb
b
C 10 == 0 not
W
> .

orthogonal

110-wll =
11 wilollvll" 21lwl 1/w// cose -
: