em disc ch1 part1 - islamic university of...
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Vector Analysis
Electrostatic
fields
Magnetostatic fields
Electromagnetic fields(wave)
Unit Vector of a vector A
is a vector whose magnitude is unity and its direction is along A
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5
___
2 2 2
2 2 2
3 4 5
| |
| |
3 4 5 50 7.071
3 4 5
7.071
0.42 0.5657 0.7071
| | 1 , , 1, , along A
A ax ay az
AaA
A
A Ax Ay Az
ax ay azaA
aA ax ay az
aA magnitude direction
Example
Position Vector of point P
is a vector directed from origin to P
e.g. OP=P-O= 2 ax + 3 ay + 4 az
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Distance Vector
directed from one point to another point .
e.g.PQ=Q-P==(0-1)ax+(3-2)ay+(1-5)az
=-ax + ay - 4 az
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azayaxazayax
PP
PPa
azayax
azayaxPPPP
PP 348.0348.087.04425
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|21|
21
225
)42()42()23(1221
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P1(2,4,4) , P2(-3,2,2)Find the unit vector along P1P2
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79436
|326|||
azayaxBA
A=ax+3azB=5ax+2ay-6azFind:
(a) |A+B|
(c) The component of A along ay
zero
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(d) Unit vector parallel to 3A+B
azayaxazayax
a
BA
azayaxBA
BA 3419.022799.091168.077
328
779464|3|
3283
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azayOROR
azayaxOPOP
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Points P(1,-3,5) , Q(2,4,6) , R(0,3,8)Find:
(a) Position vectors for P and R
(b) Distance vector QR
azayax
azayaxQRQR
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)68()43()20(
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(c) Distance between Q and R
3414||
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QR
azayaxQR
Points P(1,-3,5) , Q(2,4,6) , R(0,3,8)Find:
A=Ax.ax+Ay.ay+Az.az
B=Bx.ax+By.ay+Bz.az
A.B=|A| |B| cos θA.B=AxBx+AyBy+AzBz
Note:
A.B=-|A| |B| cos θ
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Vector Multiplication
(1) Dot Product (Scalar)
(1) If A.B=0 θAB =90 Orthogonal
ax.ay=0ax.az=0ay.az=0
(2) ax.ax=|ax|2=1
ay.ay=|ay|2=1
az.az=|az|2=1
(3) A.A=|A| |A| cos 0 =|A|2
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Notes:
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(2) Cross Product (Vector)
A=Ax.ax+Ay.ay+Az.azB=Bx.ax+By.ay+Bz.az
AxB=|A| |B| sin θ .an
AxB=
• Cross product is a vectordirection: Orthogonal to A and B planemagnitude: area of parallelogram
متوازي االضالع
(1) AxB=-(BxA)
(2) ax x ay = az
ay x az = ax
az x ax = ay
ax x az = -ay
(1) AxA=|A| |A| sin 0 = 0
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Notes:
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A=2ax-ay-2az
B=4ax+3ay+2az
(AxB).A=|AxB||A|cos 90= 0 (normal)
(AxB).B=0 (normal)العمودً علي المستوى عمودً علي أً متجه يحتويه هذا المستوى
e.g.
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A=ax+3azB=5ax+2ay-6azFind the angle between vector A and B
A.B=5+0-18=-13
65.120)5099.0(cos
5099.06510
13cos
cos||||.
6536425||
1091||
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BABA
B
A
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12210||.
2||
211||
102156)7*3()5*1()3*2(.
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2
BCA
B
B
CA
A=2ax+ay-3azB=ay-azC=3ax+5ay+7azFind:
(d) A.C - |B|2
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)4
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1( x B
2
1(d) CA
A=2ax+ay-3azB=ay-azC=3ax+5ay+7az
az)4
3ay
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19ax
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17 ( x az)
2
1-ay
2
1(
az4
3ay
12
19ax
12
17)
4
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3
1(
az4
7ay
4
5ax
4
3 C
4
1
az-ay3
1ax
3
2A
3
1
2
1-ay
2
1 B
2
1
CA
az
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az)4
3ay
12
19ax
12
17 ( x az)
2
1-ay
2
1(
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A=5ax+3ay+2az , B=-ax+4ay+6az , C=8ax+2ay Find α and β such that :αA+ βB + C is parallel to y-axis
αA+ βB + C= [ 5α ax+3α ay+ 2α az] +
[-β ax+4β ay+6β az] +[8ax+2ay]
=(5α- β+8)ax + (3α +4β +2)ay+ (2α +6β) az
5α- β+8=0 …..(1)
2α +6β=0 ……(2)
solving (1) and (2)
α=-1.5 , β=-0.5
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A= αax+ 3ay- 2az , B=4ax+ β ay+ 8 az
AxB=|A||B|sin 0 =0
(a) Find α and β if A an B are parallel?
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A= αax+ 3ay- 2az , B=4ax+ β ay+ 8 az
A.B=|A||B| cos 90 =0
4α + 3β -16 = 0
α= 0.25 – 0.75 β
(b) Relationship between α and β if A an B are perpendicular?
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(b) Show that
2 2
ay x az
ax . ay x az .
| || | cos0 | | 1
ax
ax ax
ax ax axax
ax ax ax
ay x azax
ax . ay x az
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(3) Scalar Triple Product
A.(BxC)=B.(CxA)= C.(AxB)
= volume of Parallelepipedمتوازي السطوح
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Show thatA.(BxC)=(AxB).C
