Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time,...
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Transcript of Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time,...
![Page 1: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/1.jpg)
Vector
![Page 2: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/2.jpg)
Scaler versus Vector
Scaler (向量 ): : described by magnitudeE.g. length, mass, time, speed, etc
Vector(矢量 ): described by both magnitude and directionE.g. velocity, force, acceleration, etc
Quiz: Temperature is a scaler/vector.
![Page 3: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/3.jpg)
Representing Vector
Vector can be referred to as AB or a or a
Two vectors are equal if they have the same magnitude and direction
Magnitudes equal: |a| = |c| or a = c
Direction equal: they are parallel and pointing to the same direction
A
B
AB or a
CD or c
C
D
How about these? Are they equal?a
b
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Opposite Vectors
magnitudes are equal, parallel but opposite in
sense These two vectors are not
equal Actually, they have the
relationb = -a
ab
![Page 5: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/5.jpg)
Rectangular components of Vector
A vector a can be resolved into two rectangular components or x and y components
x-component: ax
y-component: ay
a = [ax, ay]
y
x
aay
ax
Ө
![Page 6: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/6.jpg)
Addition of Vectors
V1
V2
V1
V2
V1 + V2
V1
V2
V1 + V2
Method 1Method 2
![Page 7: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/7.jpg)
Subtraction of Vectors
V1
V2
-V2
-V2
V1
V1 - V2
![Page 8: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/8.jpg)
Scaling of vectors (Multiply by a constant)
V1
V1
V1
2V1
0.5V1
-V1
![Page 9: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/9.jpg)
Class work
Given the following vectors V1 and V2. Draw on the provided graph paper:
V1+V2
V1-V2
2V1
V1
V2
![Page 10: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/10.jpg)
Class Work
For V1 given in the previous graph: X-component is _______ Y-component is _______ Magnitude is _______ Angle is _________
![Page 11: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/11.jpg)
Rectangular Form and Polar Form For the previous V1
Rectangular Form [x, y]: [4, 2]
Polar Form r Ө : √20 26.57
magnitude angle
x-component
y-component
x
yAngle
yxMagnitude
1
22
tan
![Page 12: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/12.jpg)
Polar Form Rectangular Form Since
Therefore:
Vx = |V| cos Ө
Vy = |V| sin Ө
|V|Vy
Vx
Ө
magnitude of vector V
V
V
V
V
y
x
sin
cos
![Page 13: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/13.jpg)
Example
Find the x-y components of the following vectors A, B & C
Given : |A|=2, ӨA =135o
|B|=4, ӨB = 30o
|C|=2, ӨC = 45o
y
x
A
B
C
ӨA
ӨB
ӨC
![Page 14: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/14.jpg)
Example (Cont’d)
For vector A, Ay = 2 x sin(135o) = 2, Ax = 2 x cos(135o) = -2
For vector B, Bx = 4 x cos(240o) = -4 x cos(60o) = -2,
By = 4 x sin(240o) = -4 x sin(60o) = -23
For vector C, Cy = 2 x sin(-315o) = 2 x sin(45o) = 2
Cx = 2 x cos(-315o) = 2 x cos(45o) = 2
![Page 15: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/15.jpg)
Example
What are the rectangular coordinates of the point P with polar coordinates 8π/6
Solution: use x=r sin Ө and y=r cos Ө y=8sin(π/6)=8(1/2)=4 x=8cos(π/6)=8(3 /2)=43 Hence, the rectangular coordinates are
[43,4]
π/6
8y
x
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Class work
Find the polar coordinates for the following vectors in rectangular coordinates.
V1 = [1,1] r=____ Ө=_______ V2=[-1,1] r=____ Ө=_______ V3=[-1,-1] r=____ Ө=_______ V4=[1,-1] r=____ Ө=_______
![Page 17: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/17.jpg)
Class work
a = [6, -10], r=____ Ө=_______ b = [-6, -10], r=____ Ө=______ c = [-6, 10], r=____ Ө=______ d = [6, 6], r=____ Ө=_______
![Page 18: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/18.jpg)
y
=’ x0
x
y
v
v1tan'
’’ x
y
0
'180
tan' 1
x
y
v
v
’’
x
y
0
)'180(
'180
tan' 1
x
y
v
v ’’
x
y
0
'
'360
tan' 1
x
y
v
v
oror
![Page 19: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/19.jpg)
Unit Vector A vector of length 1 unit is called a unit vector
represents a unit vector in the direction of positive x-axis represents a unit vector in the direction of positive y-axis represents a unit vector in the direction of positive z-axis Example on 2-dimensional case:
v
vvv of vector unit ˆ:
-3i
i
5i
x
y
-2j4j
j
x
y
2i
k
j
i
ˆ
ˆ
ˆ
![Page 20: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/20.jpg)
Representing a 2D vector by i and j
modulus or magnitude (length or strength) of vector
For a 3D vector represented by i, j, and kkcjbiav ˆˆˆ
y
x
|v|
a
b
i
j
22|| bav
222|| cbav
jbiav ˆˆ
![Page 21: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/21.jpg)
Vector addition If Then
Example: Given
Then
,ˆˆˆ kajaiaa zyx
y
x
a
b
bx
by
aya+b
ax
kbjbibb zyxˆˆˆ
kbajbaibaba zzyyxxˆ)(ˆ)(ˆ)(
jirjip ˆ2ˆ,ˆ3ˆ2
ji
jirp
ˆ5ˆ3
ˆ)23(ˆ)12(
![Page 22: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/22.jpg)
Vector subtraction Similarly, for
Then
Example Given
Then
kbajbaibaba zzyyxxˆ)(ˆ)(ˆ)(
y
x
a
b
by
by
ay
ax-b
y
x
aa+(-b)
,ˆˆˆ kajaiaa zyx kbjbibb zyx
ˆˆˆ
jirjip ˆ2ˆ,ˆ3ˆ2
ji
jirp
ˆˆ
ˆ)23(ˆ)12(
![Page 23: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/23.jpg)
Exercise
A = 2i + 3j, B= i j
A+B =______________AB =_______________3A =_________________|A| = ______________the modulus of B =______
![Page 24: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/24.jpg)
Example Given a=7i+2j and b=6i-5j, find a+b, a-b and modulus of a+b
Solution
jijiji ˆ3ˆ13)ˆ5ˆ6()ˆ2ˆ7(ba
jiji
jiji
ˆ7ˆˆ)52(ˆ)67(
)ˆ5ˆ6()ˆ2ˆ7(ba
178)3(13ba 22
![Page 25: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/25.jpg)
Application of force system Find the x and y components of the resultant
forces acting on the particle in the diagram
Solution:
kNjijRiRR
kNRcomponenty
kNRcomponentx
yx
y
x
ˆ035.4ˆ332.1ˆˆ
035.4)15sin(430sin6:
332.1)15cos(430cos6:00
00
6kN 4kN
15 30
y
x
![Page 26: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/26.jpg)
Scalar Product of Vectors
Scalar product, or dot product, of 2 vectors:
The result is a scalar
a
bcos|||| baba
=Angle between the 2 vectors
![Page 27: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/27.jpg)
Example:
= _________
=___________
=__________
= ___________
ii
40 degrees20 degrees
2
1
a
b
ij
ji
ba
![Page 28: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/28.jpg)
For x-y-z coordinates,
It can be shown that
zzyyxx babababa
kajaiaa zyxˆˆˆ
kbjbibb zyxˆˆˆ
![Page 29: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/29.jpg)
Example: If and Find , and angle between two
vectors Solution:
jia ˆ6ˆ4
jib ˆ3ˆ3
ba
ab
66)3(43
6)3(634
ab
ba
Notice that a b = b a
0
22
22
3.101
196.01852
6
||||cos
18])3(3[||
52)64(||
ba
ba
b
a
![Page 30: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/30.jpg)
Properties of scalar product1. Commutative:
2. Distributive:
3. For two vectors and , and a scalar C,
Example:
abba
cabacba
)(
)()()( bababa
CCC
a
b
Given A = [1,2], B=[2,-3], C=[-4,5]Find:
A(B+C) = _________AB + A C = _________3 A B = __________A (3B) = ___________
![Page 31: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/31.jpg)
If two vectors are perpendicular to each other, then their scalar product is equal to zero.
i.e. if then Example
Given and
Show that and are mutually perpendicular
Solution:
ba
0ba
a
bjia ˆ4ˆ3
jib ˆ3ˆ4
ba
jijiba
01212
)ˆ3ˆ4()ˆ4ˆ3(
![Page 32: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/32.jpg)
Vector Product of Vectors Vector product, or cross product,
Defined as
The result is a vector Magnitude = Pointing in the direction of is a unit vector perpendicular to
the plane containing and in a sense defined by the right-handed screw rule
ba
ebaba ˆsin||||
a
sin|||| ba
ee
b
sinbalength
bxa
a
b
e
![Page 33: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/33.jpg)
• Right-Hand-Rule for Cross Product
a
b
a X b
![Page 34: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/34.jpg)
Some properties of vector product If Ө=0o, then
If Ө=90o, then
It can be proven that
0ba
ebaba
abba
![Page 35: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/35.jpg)
Properties of vector product NOT commutative: Distributive: For 3D basic unit vector:
Easy way: using right-hand rule
0ˆˆ,0ˆˆ,0ˆˆ kkjjii
jik
ikj
kji
ˆˆˆ
ˆˆˆ
ˆˆˆ
abba
)()()( cabacba
i
k j
+ve-ve
jki
ijk
kij
ˆˆˆ
ˆˆˆ
ˆˆˆ
![Page 36: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/36.jpg)
For 3D vectors:
By determinant:
kbabajbabaibababa xyyxxzzxyzzyˆ)(ˆ)(ˆ)(
,ˆˆˆ kajaiaa zyx
kbjbibb zyxˆˆˆ
kbabajbabaibaba
bb
aak
bb
aaj
bb
aai
bbb
aaa
kji
ba
xyyxxzzxyzzy
yx
yx
zx
zx
zy
zy
zyx
zyx
ˆ)(ˆ)(ˆ)(
ˆˆˆ
ˆˆˆ
![Page 37: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/37.jpg)
Example:
Simplify Solution:
)ˆˆ(ˆ jij
k
k
jjijjij
ˆ
0ˆ
ˆˆˆˆ)ˆˆ(ˆ
![Page 38: Vector. Scaler versus Vector Scaler ( 向量 ): : described by magnitude E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and.](https://reader038.fdocuments.net/reader038/viewer/2022102611/56649dea5503460f94ae5211/html5/thumbnails/38.jpg)
Example: Evaluate and calculate
if and
Solution:
ba
kjia ˆ5ˆ2ˆ3
kjib ˆ8ˆ4ˆ7
|| ba
kji
kji
kji
ba
ˆ26ˆ59ˆ4
ˆ]7)2(43[ˆ]75)8(3[ˆ]45)8)(2[(
847
523
ˆˆˆ
6.644173
)26()59()4(|| 222
ba