Chapter One: Vector Analysis The use of vectors and vector analysis ...

Electromagnetic Theorem
(Dr. Omed Ghareb Abdullah) University of Sulaimani –College of Science – Physics Department
OP (1 0) aˆ
x
( 3
0) aˆ
y
(5 0) aˆ
z
aˆ
(a).
OR (0 0) aˆ
(3 0) aˆ
(8 0) aˆ
3aˆ
x
y
z
x
y
3aˆ
8aˆ
z
y
5aˆ
z
(b). r
QP
( 1
2) aˆ
( 3
4) aˆ
(5 6) aˆ
aˆ
7 aˆ
aˆ
x
y
z
x
y
z
(c).
2
2
2
QR (0 2) (3 4) (8 6) 4 1
4 3unit
Example(12):
Derive the cosine formula
c
2
2 2
a b 2abcos
and the sine formula
sin A sin B sin C
using dot and cross product respectively.
a b c
b
C
Cosine formula:
c a
b
c c (
a b)
(
a b)
a
c
2
a
2
b
2
2
b
2
A
c
2a
b
2 2
2ab
cos c a b 2abcos
a
B
Sine formula:
1 1
a b b c
2 2
1
2
a bsin
c
1
2
1
c a
2
b c sin a
1
2
c a
sin b
dividing
the above equation
by
( a b c ) we get :
sin c
c
sin a
a
sin b
b
Example(13):
Let E 3aˆ
4aˆ
and F 4aˆ
10aˆ
5aˆ
, then find:
x
z
x
y
z
(a). the component of
E along
F
(b). A unit vector perpendicular to both
E and
F
Solution:
(a). the vector component of
E along F is given by:
63