Embed Size (px)
Citation preview
Vector Components
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
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
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 ������������������������������������������
Calculation of vector components
22 )()(YXAAA
X
Y
AA
tg
cosAAX
X
YA
θ
sinAAY
xA��������������
yA��������������
PythagorasVectorModule
Example
Calculate vector A components
x
y30A
060
30 60 15cos cosXA A
30 60 15 3sin sinYA A
xA��������������
yA��������������
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
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
YYY
XXX
BAC
BAC
BAC
22
YXCCC
X
Y
A
θ
xA��������������
yA�������������� β
B
C
xB
��������������yB
��������������yC
��������������XC
Y
X
Ctg
C
Unit Vector
The unit vector is marked with a ^ signabove the letter
X
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.
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
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
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
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
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
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
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
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
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
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
