1 Vectors. 2 Vectors and Scalars, Addition of vectors Subtraction of vectors.
Addition of multiple vectors via the graphical solution is complex E A D C B It’s easier to use...
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Transcript of Addition of multiple vectors via the graphical solution is complex E A D C B It’s easier to use...
![Page 1: Addition of multiple vectors via the graphical solution is complex E A D C B It’s easier to use the mathematical method that will be described now.](https://reader038.fdocuments.in/reader038/viewer/2022110400/56649dc55503460f94ab92ff/html5/thumbnails/1.jpg)
Vector Components
![Page 2: Addition of multiple vectors via the graphical solution is complex E A D C B It’s easier to use the mathematical method that will be described now.](https://reader038.fdocuments.in/reader038/viewer/2022110400/56649dc55503460f94ab92ff/html5/thumbnails/2.jpg)
Addition of multiple vectors via the graphical solution is complex
A B C D E
E
A
DC
B
It’s easier to use the mathematical method that will be described now
![Page 3: Addition of multiple vectors via the graphical solution is complex E A D C B It’s easier to use the mathematical method that will be described now.](https://reader038.fdocuments.in/reader038/viewer/2022110400/56649dc55503460f94ab92ff/html5/thumbnails/3.jpg)
Mathematical MethodAll vectors are placed on the origin of a
Cartesian Coordinates System. Each Vector is replaced with it components
Mathematical MethodAll vectors are placed on the origin of a
Cartesian Coordinates System. Each Vector is replaced with it components
X
Y
AD
C
B
![Page 4: Addition of multiple vectors via the graphical solution is complex E A D C B It’s easier to use the mathematical method that will be described now.](https://reader038.fdocuments.in/reader038/viewer/2022110400/56649dc55503460f94ab92ff/html5/thumbnails/4.jpg)
The vector components of vector A are two perpendicular vectors Ax and Ay that are
parallel to X and Y axis respectively, add their resultant vector is equal to A.
x
y
A
xA��������������
yA��������������
x yA A A ������������������������������������������
![Page 5: Addition of multiple vectors via the graphical solution is complex E A D C B It’s easier to use the mathematical method that will be described now.](https://reader038.fdocuments.in/reader038/viewer/2022110400/56649dc55503460f94ab92ff/html5/thumbnails/5.jpg)
Calculation of vector components
22 )()(YXAAA
X
Y
AA
tg
cosAAX
X
YA
θ
sinAAY
xA��������������
yA��������������
PythagorasVectorModule
![Page 6: Addition of multiple vectors via the graphical solution is complex E A D C B It’s easier to use the mathematical method that will be described now.](https://reader038.fdocuments.in/reader038/viewer/2022110400/56649dc55503460f94ab92ff/html5/thumbnails/6.jpg)
Example
Calculate vector A components
x
y30A
060
30 60 15cos cosXA A
30 60 15 3sin sinYA A
xA��������������
yA��������������
![Page 7: Addition of multiple vectors via the graphical solution is complex E A D C B It’s easier to use the mathematical method that will be described now.](https://reader038.fdocuments.in/reader038/viewer/2022110400/56649dc55503460f94ab92ff/html5/thumbnails/7.jpg)
Vector representation depend on the chosen coordinate system (CS).
Different CS does not change the vector,but only it’s representation
x
y
x’
y’
ax
ay
ax’ay’
The vector magnitude is independent of the CS, hence we have freedom to select the CS to ease
the calculation
![Page 8: Addition of multiple vectors via the graphical solution is complex E A D C B It’s easier to use the mathematical method that will be described now.](https://reader038.fdocuments.in/reader038/viewer/2022110400/56649dc55503460f94ab92ff/html5/thumbnails/8.jpg)
Vector addition, is done by adding the respective components
Sum of the respective components yield the resultant vector components,
hence we can find the resultant vector
X
Y
A
θ
xA��������������
yA�������������� β
B
C
C A B
xB
��������������yB
![Page 9: Addition of multiple vectors via the graphical solution is complex E A D C B It’s easier to use the mathematical method that will be described now.](https://reader038.fdocuments.in/reader038/viewer/2022110400/56649dc55503460f94ab92ff/html5/thumbnails/9.jpg)
YYY
XXX
BAC
BAC
BAC
22
YXCCC
X
Y
A
θ
xA��������������
yA�������������� β
B
C
xB
��������������yB
��������������yC
��������������XC
Y
X
Ctg
C
![Page 10: Addition of multiple vectors via the graphical solution is complex E A D C B It’s easier to use the mathematical method that will be described now.](https://reader038.fdocuments.in/reader038/viewer/2022110400/56649dc55503460f94ab92ff/html5/thumbnails/10.jpg)
Unit Vector
The unit vector is marked with a ^ signabove the letter
X
![Page 11: Addition of multiple vectors via the graphical solution is complex E A D C B It’s easier to use the mathematical method that will be described now.](https://reader038.fdocuments.in/reader038/viewer/2022110400/56649dc55503460f94ab92ff/html5/thumbnails/11.jpg)
Unit vector is a vector the has a magnitude of 1 with no units, and has a direction
It is described by it’s components as any vector.
![Page 12: Addition of multiple vectors via the graphical solution is complex E A D C B It’s easier to use the mathematical method that will be described now.](https://reader038.fdocuments.in/reader038/viewer/2022110400/56649dc55503460f94ab92ff/html5/thumbnails/12.jpg)
The CS unit vectors are unit vectors pointing parallel to the X,Y and Z axes: x, y, z
x
y
z
x
y
z
Representing a vector by it’s components is as follows:
ˆ ˆ ˆ ��������������
X y zA A x A y A z
![Page 13: Addition of multiple vectors via the graphical solution is complex E A D C B It’s easier to use the mathematical method that will be described now.](https://reader038.fdocuments.in/reader038/viewer/2022110400/56649dc55503460f94ab92ff/html5/thumbnails/13.jpg)
Determine unit vector components of A
ˆ ˆ ˆ8 6 10A x y z
Draw the vector on the CS
x
y
z
8x
6 ˆ y
10z
A
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ˆ ˆ ˆ8 6 10A x y z
Magnitude of A
x
y
z
8x
6 ˆ y
10z
2 2 28 6 10 14.14A A
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ˆ ˆ ˆ8 6 10ˆ ˆ ˆ ˆ0.56 0.42 0.714.14
A x y zA x y z
A
Divide each component by the vector magnitude, results in a unit vector having same direction a vector A
x
y
z
8x
6 ˆ y
10zA
![Page 16: Addition of multiple vectors via the graphical solution is complex E A D C B It’s easier to use the mathematical method that will be described now.](https://reader038.fdocuments.in/reader038/viewer/2022110400/56649dc55503460f94ab92ff/html5/thumbnails/16.jpg)
x
y
3 cm
A
B6 cm
450
Calculate:1.The components of vectors A and B2.The resultant vector components3.The resultant vector magnitude and direction4.Find the unit vector parallel to the resultant
vector5.A vector with magnitude 10 that is parallel to
the resultant vector
![Page 17: Addition of multiple vectors via the graphical solution is complex E A D C B It’s easier to use the mathematical method that will be described now.](https://reader038.fdocuments.in/reader038/viewer/2022110400/56649dc55503460f94ab92ff/html5/thumbnails/17.jpg)
3 cmx
y
A
B 6 cm
450
The components of vectors A and B
0 0
3 0
6 45 6 45 4 2 4 2
ˆ ˆ
cos ˆ sin ˆ . ˆ . ˆ
A x y
B x y x y
![Page 18: Addition of multiple vectors via the graphical solution is complex E A D C B It’s easier to use the mathematical method that will be described now.](https://reader038.fdocuments.in/reader038/viewer/2022110400/56649dc55503460f94ab92ff/html5/thumbnails/18.jpg)
3 cmx
y
A
B 6 cm
450
The resultant vector components
3 0
4 2 4 2
ˆ ˆ
. ˆ . ˆ
A x y
B x y
3 4 2 0 4 2
1 2 4 2
( . ) ˆ ( . )ˆ
. ˆ . ˆ
C x y
C x y
C A B
![Page 19: Addition of multiple vectors via the graphical solution is complex E A D C B It’s easier to use the mathematical method that will be described now.](https://reader038.fdocuments.in/reader038/viewer/2022110400/56649dc55503460f94ab92ff/html5/thumbnails/19.jpg)
The resultant vector magnitude and direction
x
y
2 2
0
1 2 4 2 4 37
4 23 5
1 2
74 05
. . .
..
.
.
C cm
tg
4.37
-1.2
740
4.2
Above Negative X Axis
C= 1 2 4 2. ˆ . ˆC x y
![Page 20: Addition of multiple vectors via the graphical solution is complex E A D C B It’s easier to use the mathematical method that will be described now.](https://reader038.fdocuments.in/reader038/viewer/2022110400/56649dc55503460f94ab92ff/html5/thumbnails/20.jpg)
x
y
4.37
-1.2
740
4.2C= 1 2 4 2. ˆ . ˆC x y
Find the unit vector parallel to the resultant vector
1 2 4 20 27 0 96
4 37
. ˆ . ˆˆ . ˆ . ˆ.
C x yC x y
C
![Page 21: Addition of multiple vectors via the graphical solution is complex E A D C B It’s easier to use the mathematical method that will be described now.](https://reader038.fdocuments.in/reader038/viewer/2022110400/56649dc55503460f94ab92ff/html5/thumbnails/21.jpg)
A vector with magnitude 10 that is parallel to the resultant vector
0 27 0 96ˆ . ˆ . ˆC x y
10 10 0 27 0 96 2 7 9 6ˆ ( . ˆ . ˆ) . ˆ . ˆD C x y x y
Unit vector parallel to resultant vector
![Page 22: Addition of multiple vectors via the graphical solution is complex E A D C B It’s easier to use the mathematical method that will be described now.](https://reader038.fdocuments.in/reader038/viewer/2022110400/56649dc55503460f94ab92ff/html5/thumbnails/22.jpg)