1638 vector quantities
TRANSCRIPT
Fall 2006Fall 2006
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- Cylindrical3- Spherical
Specified by one of the following coordinates best applied to application:
Page 101
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
Tvvvvvvvvv
321321
3
2
1
v
Unit vector
Magnitude
222
222
ˆˆˆˆ
ˆˆˆ
zyx
zyx
zyx
zyx
AAA
AzAyAxAAa
AAAAA
AzAyAxA
Page 101-102
a
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
vv
vv
Pv
both vectors equal
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.
5.07.05.0
v
Cartesian Coordinate System
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.
22221 n
vvv v
11131131
222
Normal Vectors
• When we wish to describe direction we use normalized vectors.
• We often need to normalize a vector:
vvvv
22221
1
nvvv
Vector Addition and SubtractionVector Addition and Subtraction
)(ˆ)(ˆ)(ˆˆˆˆ
ˆˆˆˆ
ˆˆˆˆ
zzyyxx
zyx
zyx
BAzBAyBAxBbAaBACcC
BzByBxBbB
AzAyAxAaA
xC yC zC
Addition
Subtraction
212
212
21212
12121212
122112
)()()(
)(ˆ)(ˆ)(ˆ
zzyyxxR
zzzyyyxxxR
RRPPR
R12
Page 103
Vector Addition
u+v
y
x
v
u
Vector Addition
• Addition of vectors follows the parallelogram law in 2D and the parallelepiped law in higher dimensions:
vuvu
975
654
321
Vector Subtraction
y
x
v
u v-u
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
Problem 3-1
20ˆ11ˆ12ˆ zyxA
y
x
z
2,1,11 P
0,1,22 P
A
20ˆ11ˆ12ˆ zyxA
2ˆˆ zxA
24.2541 A
89.0ˆ45.0ˆ24.2
2ˆˆˆ zxzxAAa
Vector MultiplicationVector Multiplication1- Simple Product2- Scalar or Dot Product3- Vector or Cross Product
1- Simple Product
)(ˆ)(ˆ)(ˆˆ zyx kAzkAykAxkAaAkB
Scalar or Dot Product
ABABBA cos
Page 104
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):
Dot Product• Dot product (inner product) is defined as:
332211
3
2
1
321 vuvuvuvvv
uuuT
vuvu
i
iivuvu
cosvuvu
Dot Product• Note:
• Therefore we can redefine magnitude in terms of the dot-product operator:
• Dot product operator is commutative & distributive.
223
22
21 uuu uuu
uuu
Problem 3.2
• Given vectors:
– show that is perpendicular to both and .
A
B
C
zyxA ˆ3ˆ2ˆ
3ˆˆ2ˆ zyxB
2ˆ2ˆ4ˆ zyxC
Recall
10cosˆˆ
10cosˆˆ
10cosˆˆ
zz
yy
xx
zz
yy
xx
0ˆˆˆˆˆˆ zyzxyx
Also, recall
Similarly
0 BA
If the angle between the two vectors is 90.
)(ˆ)(ˆ)(ˆ xyyxzxxzyzzy BABAzBABAyBABAxBA
Show
zyx AAA zyxA ˆˆˆ
zyx BBB zyxB ˆˆˆ
yxzxzyzyx
ˆˆˆˆˆˆˆˆˆ
yzxxyzzxy
ˆˆˆˆˆˆˆˆˆ
Recall
xyyxzxxzyzzy
yzxzzyxyzxyx
zyxzyx
BABABABABABA
BABABABABABA
BBBAAA
zyx
xyxzyz
zyxzyxBA
ˆˆˆ
ˆˆˆˆˆˆ
ˆˆˆˆˆˆ
)(ˆ)(ˆ)(ˆ xyyxzxxzyzzy BABAzBABAyBABAxBA
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.
y
x
z
y
x
z
y
x
z
y
x
z 2,2,01P
y
x
z 2,2,01P
y
x
z 2,2,01P
y
x
z 2,2,01P
y
x
z 2,2,01P
y
x
z 2,2,01P 2,2,22P
y
x
z 2,2,01P 2,2,22P
y
x
z 2,2,01P 2,2,22P
y
x
z 2,2,01P 2,2,22P
y
x
z 2,2,01P 2,2,22P
y
x
z 2,2,01P 2,2,22P
2,1,13 P
y
x
z 2,2,01P 2,2,22P
2,1,13 P
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.
91821324
21464256
212816
21
ˆ2ˆ8162116ˆ4ˆ8ˆ2
21
ˆˆ16ˆˆ4ˆˆ4ˆˆ8ˆˆ2ˆˆ221
4ˆˆˆ4ˆ2ˆ21
21
222
zyzyz
zyyyxyzxyxxx
zyxyxCB
xx
A
0 z y z 0 x
• The area of the triangle is 9 sq. units.
y
x
z
0,4.63,24.2
0,107.1,51
radP
r
51422 yxr
radxy 107.1
12tantan 11
0h
y
x
z
4.63,90,24.2
107.1,57.1,51
radradP
r
5014222 zyxr
radxy 107.1
12tantan 11
0h
radzyx
57.105tantan 1
221
Convert the coordinates of P1(1,2,0) from the Cartesian to the Spherical coordinates.
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
2h
radx
785.011tan1tan 11
y
x
z
Convert the coordinates of P3(1,1,2) from the Cartesian to the Cylindrical coordinates.
2,
4,23 radP
21122 yxr
2h
radx
785.011tan1tan 11
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
rad616.022tan
211tan 1
221
rad4
1tan11tan 11