Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”)

1. Fall 2006Fall 2006

2. Scalar Quantity
(mass, speed, voltage, current and power)
1- Real number (one variable)
2- Complex number (two variables)
Vector Algebra
(velocity, electric field and magnetic field)
Magnitude & direction
1- Cartesian (rectangular)
2- Cylindrical
3- Spherical
Specified by one of the following
coordinates best applied to application:
Page 101

3. Conventions
• Vector quantities denoted as or v
• We will use column format vectors:
• Each vector is defined with respect to a set of
basis vectors (which define a co-ordinate
system).
v

[ ] [ ]( )T
vvvvvv
v
v
v
321321
3
2
1
=≠










=v

4. Unit vector
Magnitude
+
+
++
++
==
++==
++=
222
222
ˆˆˆ
ˆ
ˆˆˆ
zyx
zyx
zyx
zyx
AAA
AzAyAx
A
A
a
AAAAA
AzAyAxA



Page 101-102
aˆ

5. Vectors
• Vectors represent directions
– points represent positions
• Both are meaningless
without reference to a
coordinate system
– vectors require a set of basis
vectors
– points require an origin and a
vector space






=





=
y
x
y
x
v
v
v
v
Pv
both vectors equal

6. Co-ordinate Systems
• Until now you have probably used a Cartesian
basis:
– basis vectors are mutually orthogonal and
unit length
– basis vectors named x, y and z
• We need to define the relationship between the
3 vectors.

7. 









=
5.0
7.0
5.0
v
Cartesian Coordinate System

8. Vector Magnitude
• The magnitude or norm of a vector of
dimension n is given by the standard
Euclidean distance metric:
• For example:
• Vectors of length 1(unit) are normal vectors.
222
21 n
vvv +++= v
11131
1
3
1
222
=++=

9. Normal Vectors
• When we wish to describe direction we
use normalized vectors.
• We often need to normalize a vector:
v
v
v
v
222
21
1
n
vvv +++
==′


10. Vector Addition and SubtractionVector Addition and Subtraction
)(ˆ)(ˆ)(ˆˆˆˆ
ˆˆˆˆ
ˆˆˆˆ
zzyyxx
zyx
zyx
BAzBAyBAxBbAaBACcC
BzByBxBbB
AzAyAxAaA
±+±+±=±=±==
++==
++==



xC yC zC
Addition
Subtraction
2
12
2
12
2
1212
12121212
122112
)()()(
)(ˆ)(ˆ)(ˆ
zzyyxxR
zzzyyyxxxR
RRPPR
−+−+−=
−+−+−=
−==



R12
Page 103

11. Vector Addition
u+v
y
x
v
u

12. Vector Addition
• Addition of vectors follows the parallelogram law
in 2D and the parallelepiped law in higher
dimensions:
vuvu +










=










+










9
7
5
6
5
4
3
2
1

13. Vector Subtraction
y
x
v
u v-u

14. Problem 3-1
• Vector starts at point (1,-1,-2) and ends
at point (2,-1,0). Find a unit vector in the
direction of .
A

A

y
x
z
( ) ( )( ) ( )( )20ˆ11ˆ12ˆ −−+−−−+−= zyxA


15. Problem 3-1
( ) ( )( ) ( )( )20ˆ11ˆ12ˆ −−+−−−+−= zyxA

y
x
z
( )2,1,11 −−P
( )0,1,22 −P
A


16. ( ) ( )( ) ( )( )20ˆ11ˆ12ˆ −−+−−−+−= zyxA

2ˆˆ zxA +=

24.2541 ==+=A

89.0ˆ45.0ˆ
24.2
2ˆˆ
ˆ zx
zx
A
A
a +=
+
== 


17. Vector MultiplicationVector Multiplication
1- Simple Product
2- Scalar or Dot Product
3- Vector or Cross Product
1- Simple Product
)(ˆ)(ˆ)(ˆˆ zyx kAzkAykAxkAaAkB ++===

Scalar or Dot Product
ABABBA Θ=⋅ cos

Page 104

18. Vector Multiplication by a Scalar
• Each vector has an associated length
• Multiplication by a scalar scales the vectors
length appropriately (but does not affect
direction):

19. Dot Product
• Dot product (inner product) is defined as:
[ ] 332211
3
2
1
321 vuvuvu
v
v
v
uuuT
++=










==⋅ vuvu
∑=⋅
i
iivuvu
θcosvuvu =⋅

20. Dot Product
• Note:
• Therefore we can redefine magnitude in terms
of the dot-product operator:
• Dot product operator is commutative &
distributive.
22
3
2
2
2
1 uuu =++=⋅ uuu
uuu ⋅=

21. Problem 3.2
• Given vectors:
– show that is perpendicular to both
and .
A

B

C

zyxA ˆ3ˆ2ˆ +−=

3ˆˆ2ˆ zyxB +−=

2ˆ2ˆ4ˆ zyxC −+=


22. Recall
10cosˆˆ
10cosˆˆ
10cosˆˆ
==•
==•
==•



zz
yy
xx
zz
yy
xx
0ˆˆˆˆˆˆ =•=•=• zyzxyx
Also, recall

23. Similarly
0=• BA

If the angle between the two vectors is 90°.

24. )(ˆ)(ˆ)(ˆ xyyxzxxzyzzy BABAzBABAyBABAxBA −+−+−=×

Show
zyx AAA zyxA ˆˆˆ ++=

zyx BBB zyxB ˆˆˆ ++=


25. yxz
xzy
zyx
ˆˆˆ
ˆˆˆ
ˆˆˆ
=×
=×
=×
yzx
xyz
zxy
ˆˆˆ
ˆˆˆ
ˆˆˆ
−=×
−=×
−=×
Recall

26. ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )xyyxzxxzyzzy
yzxzzyxyzxyx
zyxzyx
BABABABABABA
BABABABABABA
BBBAAA
−+−+−=
−++−−=
++×++=×
zyx
xyxzyz
zyxzyxBA
ˆˆˆ
ˆˆˆˆˆˆ
ˆˆˆˆˆˆ

)(ˆ)(ˆ)(ˆ xyyxzxxzyzzy BABAzBABAyBABAxBA −+−+−=×


27. Problem 3.3
• In the Cartesian coordinate system, the
three corners of a triangle are P1(0,2,2),
P2(2,-2,2), and P3(1,1,-2). Find the area
of the triangle.

28. y
x
z

29. y
x
z

30. y
x
z

31. y
x
z
( )2,2,01P

32. y
x
z
( )2,2,01P

33. y
x
z
( )2,2,01P

34. y
x
z
( )2,2,01P

35. y
x
z
( )2,2,01P

36. y
x
z
( )2,2,01P
( )2,2,22P

37. y
x
z
( )2,2,01P
( )2,2,22P

38. y
x
z
( )2,2,01P
( )2,2,22P

39. y
x
z
( )2,2,01P
( )2,2,22P

40. y
x
z
( )2,2,01P
( )2,2,22P

41. y
x
z
( )2,2,01P
( )2,2,22P
( )2,1,13 −P

42. y
x
z
( )2,2,01P
( )2,2,22P
( )2,1,13 −P

43. Let
4ˆ2ˆ21 yxB −== PP

and
4ˆˆˆ31 zyxC −−== PP

→
→
represent two sides of the triangle.
Since the magnitude of the cross product is the
area of the parallelogram, half of this magnitude
is the area of the triangle.

44. ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) 918
2
1
324
2
1
464256
2
1
2816
2
1
ˆ2ˆ816
2
1
16ˆ4ˆ8ˆ2
2
1
ˆˆ16ˆˆ4ˆˆ4ˆˆ8ˆˆ2ˆˆ2
2
1
4ˆˆˆ4ˆ2ˆ
2
1
2
1
222
===++=++=
+++=+++−=
×+×+×−×−×−×=
−−×−=×=
zyzyz
zyyyxyzxyxxx
zyxyxCB
xx
A

0 zˆ yˆ− zˆ− 0 xˆ

45. • The area of the triangle is 9 sq. units.

46. y
x
z
( )
( )0,4.63,24.2
0,107.1,51

≈
radP
φ
r
51422
=+=+= yxr
rad
x
y
107.1
1
2
tantan 11
=





=





= −−
φ
0=h

47. y
x
z
( )
( )
4.63,90,24.2
107.1,57.1,51
≈
radradP
φ
r
5014222
=++=++= zyxr
rad
x
y
107.1
1
2
tantan 11
=





=





= −−
φ
0=h
rad
z
yx
57.1
0
5
tantan 1
22
1
=








=







 +
= −−
θ
Convert the coordinates of
P1(1,2,0) from the
Cartesian to the Spherical
coordinates.

48. y
x
z
Convert the coordinates of P3(1,1,2) from the Cartesian to
the Cylindrical and Spherical coordinates.
( )2,1,11P
21122
=+=+= yxr
2=h
rad
x
785.0
1
1
tan
1
tan 11
=





=





= −−
φ

49. y
x
z
Convert the coordinates of P3(1,1,2) from the Cartesian to
the Cylindrical coordinates.






2,
4
,23 radP
π
21122
=+=+= yxr
2=h
rad
x
785.0
1
1
tan
1
tan 11
=





=





= −−
φ

50. y
x
z
Convert the coordinates of P3(1,1,2) from the Cartesian to
the Spherical coordinates.
( )
45,3.35,45.23P
6211 222222
=++=++= zyxr
rad616.0
2
2
tan
2
11
tan 1
22
1
=







=







 +
= −−
θ
( ) rad
4
1tan
1
1
tan 11 π
φ ==





= −−