, Ch1 4- Vectors .
didorcet encsredr office has MWF 11-12 : 30 :
M 1/9/2is
& Riddick Hearth
Ex
0 out
in vectors :
position rector-location ↑ <4
↳0
·
= -2 03
,
,
Y
-
F =
<Vx , y z] ,
·
can also represent velocity ,
forces , etc
.
vectors have length & direction
Vector math
·
scalar multiplication
·
Ea = <4 - 2 07 , ,
= <2
,
-
1
,
0
-
ney ,
changes direction # ,
changes length
·
addition
- 73 3 5 7-3 3
t "tip
6)
a -2 07 < 1 1,
=
, ,
03
+ =
<4 , ,
+
, ,
03 =
, to tail"
·
subtraction
a -
3 = a + ( 35 = (4 , -2
,
07 + 6)7
-
< 3 ,
,
-3
7 ,%
,
07
=, 50
mini
Vector magnitude :
" 11 = c =T
jal = : To
Vector Equality :
· 2 rectors are
equal if :
>
-
components are equal
a
· = 5 if ax = bx ay by , az bz
= =
,
>
-
magnitude & direction are ali
equal
Unit Vectors ;
magnitude =
colT
·
(
= lat
↑==
l add
2x : a = <4 , - 2, 02 l =0 =
to 4 ,2 02 : 0) ,
,
ll = = 1
,Ch1 1-1 4 . .
Fri : 1 5
.
-
1 7 .
physicstrtorial center .
Wordpress ,
nasu . edu W 1/11/23
Vector Direction & Angles :
·
sin() & cos() Example :
unit rectors ↓ flat board
2020 s
2 <08x 2018y is
·
= on
- , ,
#
& 60 °
m/ -
X-axis
x
8x =
150 -
60 = 120 On
°
= 90 60-
=
300
perpendicular
·
2 02 = 90 >
- to "board"
Inertia: & =
<LOS 120 ,
10338 10s907 ,
8 = < - 0 5 .
,
0 . 866 , 03
F 1/13/23
v ( 9, A , -20 > = < 0 36 , 48,
/V
0
B) =? ?
=
A
=?
=
B
.
.
=
↑ 9
,
7,
10) 8 = , , - Pers
↑ = 0 .
36
oth = /r) = 25
= = 0 . 48 A = 12
Ch1 5- 1 . .
7
Velocity
in
displacement (change
② >
- 1 2 @F = -
F
T
o
from of
position)
v=
·
- =
or
average velocity =
placement
~ = (in-5) Targ
arg Ot =*-* ,
positionupdate =+ Fae
ball is ay < 1
moving w/
A 2 , 37 m
ex : ,
,
constant velocity of 24 ,
2 03
,
/S . Location & t = 35
= = < 1
,
2
,
33 + (4 ,
2
,
03 ·
3) = <, 2
,
37 + 212 ,
6
, 07 = 413
,
8, 33u
< 4 ,
3 27 met
,
=
6s &t = lls <14 - 2, 27 m Varg :?
Varg
= = 3
,
2-2 50) = 2, 0
, Ch 1 5- 1 . .
7 Cont .
F 1/13/23
Changing Speed :
t = 1) = <
3
(m) Varg (t 0 to 3 45 <2, 4) <1 , 0 0 m/s
I
t(s) ↓ = , 3 ,
-
, =
,
7
O <2, 3 , 47 Varg (t 0 to =
+= 2) =
4 4) (2 3 47
<1 50 m/s
-
, , , , = .
,
02
1 <3 , 3 , 4)
2 < 5, 3, % Varg(t = 0 tot = 3) = <2 ,
0
,
0 > M/s
348 3 , ,
44
call
Stuff [varg : = = (*) , (y) , (2)) =
Vinst]
ex :(t) = < t2 , -5t 3 > m
,
Vars from t = 1 to - = 2 :
(t 1) < (174 511) 3)
=
<1 , 5 37
-
= = -
,
M
, ,
↑ (t 2) = =
22,
-
5/2) 3) = <4 , -10 37M
Varg < 4 , -10 , 33 <1 -5, 3) ,
-
,
= ,
= <3 , -
5, 07 m/s
VinstH 4) =
= (r)( =
<(2) EE5A 813K le , , : 1
=
= 2
< 27
-
5 0 >
et = 2 = < 4 -5 0x M/s
,
,
, ,
↑ < 2, -3 t +2) @t = 2 find angle bl &+ axis
s Vinst(t , St),
= v z
, ,
= 2) =
<(2) / + >k = = 0, -
3
,
4)
n ↑ ~ = +
Winst
v =
5 = 10 ,0 6 , 0 .
.
87
- ↓ 0 .
U = cos
10 8)
O2
Z
5 arcos . =
E2
tant = 36 870
13 0 = .
Acceleration :
= tinst) = tl))
5
· i = cost) ,
+ ,
37
-
Vinst =< -Sin(t) 57% ,
07
,
a = <
10s(1) 20t" ,
,
0 >
didorcet encsredr office has MWF 11-12 : 30 :
M 1/9/2is
& Riddick Hearth
Ex
0 out
in vectors :
position rector-location ↑ <4
↳0
·
= -2 03
,
,
Y
-
F =
<Vx , y z] ,
·
can also represent velocity ,
forces , etc
.
vectors have length & direction
Vector math
·
scalar multiplication
·
Ea = <4 - 2 07 , ,
= <2
,
-
1
,
0
-
ney ,
changes direction # ,
changes length
·
addition
- 73 3 5 7-3 3
t "tip
6)
a -2 07 < 1 1,
=
, ,
03
+ =
<4 , ,
+
, ,
03 =
, to tail"
·
subtraction
a -
3 = a + ( 35 = (4 , -2
,
07 + 6)7
-
< 3 ,
,
-3
7 ,%
,
07
=, 50
mini
Vector magnitude :
" 11 = c =T
jal = : To
Vector Equality :
· 2 rectors are
equal if :
>
-
components are equal
a
· = 5 if ax = bx ay by , az bz
= =
,
>
-
magnitude & direction are ali
equal
Unit Vectors ;
magnitude =
colT
·
(
= lat
↑==
l add
2x : a = <4 , - 2, 02 l =0 =
to 4 ,2 02 : 0) ,
,
ll = = 1
,Ch1 1-1 4 . .
Fri : 1 5
.
-
1 7 .
physicstrtorial center .
Wordpress ,
nasu . edu W 1/11/23
Vector Direction & Angles :
·
sin() & cos() Example :
unit rectors ↓ flat board
2020 s
2 <08x 2018y is
·
= on
- , ,
#
& 60 °
m/ -
X-axis
x
8x =
150 -
60 = 120 On
°
= 90 60-
=
300
perpendicular
·
2 02 = 90 >
- to "board"
Inertia: & =
<LOS 120 ,
10338 10s907 ,
8 = < - 0 5 .
,
0 . 866 , 03
F 1/13/23
v ( 9, A , -20 > = < 0 36 , 48,
/V
0
B) =? ?
=
A
=?
=
B
.
.
=
↑ 9
,
7,
10) 8 = , , - Pers
↑ = 0 .
36
oth = /r) = 25
= = 0 . 48 A = 12
Ch1 5- 1 . .
7
Velocity
in
displacement (change
② >
- 1 2 @F = -
F
T
o
from of
position)
v=
·
- =
or
average velocity =
placement
~ = (in-5) Targ
arg Ot =*-* ,
positionupdate =+ Fae
ball is ay < 1
moving w/
A 2 , 37 m
ex : ,
,
constant velocity of 24 ,
2 03
,
/S . Location & t = 35
= = < 1
,
2
,
33 + (4 ,
2
,
03 ·
3) = <, 2
,
37 + 212 ,
6
, 07 = 413
,
8, 33u
< 4 ,
3 27 met
,
=
6s &t = lls <14 - 2, 27 m Varg :?
Varg
= = 3
,
2-2 50) = 2, 0
, Ch 1 5- 1 . .
7 Cont .
F 1/13/23
Changing Speed :
t = 1) = <
3
(m) Varg (t 0 to 3 45 <2, 4) <1 , 0 0 m/s
I
t(s) ↓ = , 3 ,
-
, =
,
7
O <2, 3 , 47 Varg (t 0 to =
+= 2) =
4 4) (2 3 47
<1 50 m/s
-
, , , , = .
,
02
1 <3 , 3 , 4)
2 < 5, 3, % Varg(t = 0 tot = 3) = <2 ,
0
,
0 > M/s
348 3 , ,
44
call
Stuff [varg : = = (*) , (y) , (2)) =
Vinst]
ex :(t) = < t2 , -5t 3 > m
,
Vars from t = 1 to - = 2 :
(t 1) < (174 511) 3)
=
<1 , 5 37
-
= = -
,
M
, ,
↑ (t 2) = =
22,
-
5/2) 3) = <4 , -10 37M
Varg < 4 , -10 , 33 <1 -5, 3) ,
-
,
= ,
= <3 , -
5, 07 m/s
VinstH 4) =
= (r)( =
<(2) EE5A 813K le , , : 1
=
= 2
< 27
-
5 0 >
et = 2 = < 4 -5 0x M/s
,
,
, ,
↑ < 2, -3 t +2) @t = 2 find angle bl &+ axis
s Vinst(t , St),
= v z
, ,
= 2) =
<(2) / + >k = = 0, -
3
,
4)
n ↑ ~ = +
Winst
v =
5 = 10 ,0 6 , 0 .
.
87
- ↓ 0 .
U = cos
10 8)
O2
Z
5 arcos . =
E2
tant = 36 870
13 0 = .
Acceleration :
= tinst) = tl))
5
· i = cost) ,
+ ,
37
-
Vinst =< -Sin(t) 57% ,
07
,
a = <
10s(1) 20t" ,
,
0 >
