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Review: Analysis vector
VECTOR ANALYSIS
1.1 SCALARS AND VECTORS
1.2 VECTOR COMPONENTS AND UNIT VECTOR
1.3 VECTOR ALGEBRA
1.4 POSITION AND DISTANCE VECTOR
1.5 SCALAR AND VECTOR PRODUCT OF
VECTORS
• A scalar quantity – has only magnitude• A vector quantity – has both magnitude
and direction
1.1 SCALARS & VECTORS
electric field intensity
• A vector in Cartesian Coordinates maybe represented as
zyx RRR ,,R
1.2 VECTOR COMPONENTS & UNIT VECTOR
R
Or
zzyyxx RRR aaaR
R
yRThe vector has three component vectors, which are , and
zRxR
VECTOR COMPONENTS & UNIT VECTOR (Cont’d)
• Each component vectors have magnitude
which depend on the given vector and they
have a known and constant direction.
• A unit vector along is defined as a
vector whose magnitude is unity and
directed along the coordinate axes in the
direction of the increasing coordinate
values
R
VECTOR COMPONENTS & UNIT VECTOR (Cont’d)
Any vector maybe described asR
zzyyxx RRR aaaR
The magnitude of written or simply given by
R R
R
222zyx RRR R
VECTOR COMPONENTS & UNIT VECTOR (Cont’d)
Unit vector in the direction of the vector is:
R
R
RRa
222zyx
RRRR
VECTOR COMPONENTS & UNIT VECTOR (Cont’d)
EXAMPLE 1
Specify the unit vector extending from
the origin toward the point
1,2,2 G
SOLUTION TO EXAMPLE 1
Construct the vector extending from origin to point G
Find the magnitude of
zyx aaaG
22G
3122 222 G
So, unit vector is:
zyx
zyxG
aaa
aaaG
Ga
333.0667.0667.0
3
1
3
2
3
2
SOLUTION TO EXAMPLE 1 (Cont’d)
1.3 VECTOR ALGEBRA
• Two vectors, and can be added together to give another vector
A
B
C
BAC
• Let zyx AAA ,,A zyx BBB ,,
B
zzzyyyxxx BABABA aaaC
VECTOR ALGEBRA (Cont’d)
Vectors in 2 components
• Vector subtraction is similarly carried out as:
)B(ABAD
zzzyyyxxx BABABA aaaD
VECTOR ALGEBRA (Cont’d)
VECTOR ALGEBRA (Cont’d)
• Laws of Vectors:
Associative Law
Commutative Law
Distributive Law
Multiplication by Scalar
CB)(AC)(BA
ABBA
BAB)(A aaa
BABAB)(AB)(AB)(A ssrrsrsr )(
EXAMPLE 2
If
Find: (a) The component of along
(b) The magnitude of
(c) A unit vector along
zyx aaaA 6410
yx aaB
2
A ya
BA3
BA 2
(a) The component of along is
A ya
4yA
(b)
zyx aaa
BA
181328
18,13,28
0,1,218,12,30
0,1,26,4,1033
SOLUTION TO EXAMPLE 2
Hence, the magnitude of is:
BA3
74.351813283 222 BA
(c) Let
zyx aaa
B2AC
6214
6,2,14
0,2,46,4,10
SOLUTION TO EXAMPLE 2 (Cont’d)
zyx
zyx
C
aaa
aaa
C
Ca
391.0130.0911.036.15
6
36.15
2
36.15
14
6214
6,2,14222
So, the unit vector along is:C
SOLUTION TO EXAMPLE 2 (Cont’d)
• A point P in Cartesian coordinate maybe represented as
• The position vector (radius vector) of point P is as the directed distance from the origin O to point P is
zyxP zyxOP aaar
zyx ,,P
Pr
1.4 POSITION AND DISTANCE VECTOR
zyxP aaar 543
POSITION AND DISTANCE VECTOR (Cont’d)
POSITION AND DISTANCE VECTOR (Cont’d)
• If we have two
position vectors,
and , the third
vector or “distance
vector” can be defined
as:
Pr
Qr
PQPQ rrr
Point P and Q are located at
and . Calculate:
4,2,0 5,1,3
(a) The position vector P
(b) The distance vector from P to Q
(c) The distance between P and Q
(d) A vector parallel to with magnitude
of 10
PQ
EXAMPLE 3
(a)
(b)
(c)
zyzyxP aaaaar 42420
zyx
PQPQ
aaa
rrr
3
4,2,05,1,3
Since is the distance vector, the distance between P and Q is the magnitude of this distance vector.
PQr
SOLUTION TO EXAMPLE 3
SOLUTION TO EXAMPLE 3 (Cont’d)
Distance, d
317.3113 222 PQd r
(d) Let the required vector be then
Where is the magnitude of
AAaA
A
10AA
Since is parallel to , it must
have same unit vector as or
A
PQ
SOLUTION TO EXAMPLE 3 (Cont’d)
PQr
QPr
317.3
1,1,3
PQ
PQAa
r
r
So, 317.3
1,1,310
A
cos ( , )
cosAB
ABAB
A B A B A B
cos ABB
AB
A
B
cos ABA
ABEnclosed Angle
cos
cosBA
AB
BA
AB
A B B A
cos cosAB AB
cos
arccos
AB
AB
A B
A B
A B
A B
SCALAR PRODUCT OF VECTORS
1.5 SCALAR AND VECTOR PRODUCT OF VECTORS
sin ( , )
sin
AB
AB
AB
C
AB
S
C A×B
A B A B
AB
A
B
C
ABS
and /
und C A C B
Surface
and
VECTOR PRODUCT OF VECTORS
Add the first two Columns
( )
+ ( )
( )
x y z
x y z
x y z
x y z x y
x y z x y
x y z x y
y z z y x
z x x z y
x y y x z
A A A
B B B
A A A A A
B B B B B
A B A B
A B A B
A B A B
e e e
A×B
e e e e e
e
e
e
Sarrus Law[Pierre Frédéric Sarrus, 1831]
http://de.wikipedia.org/wiki/Regel_von_Sarrus
VECTOR PRODUCT OF VECTORS (Cont’d)
Properties of cross product of unit vectors:
yxzxzyzyx aaaaaaaaa ,,
Or by using cyclic permutation:
VECTOR PRODUCT OF VECTORS (Cont’d)
Determine the dot product and cross product of the following vectors:
zyx
zyx
aaaB
aaaA
65
432
EXAMPLE 4
The dot product is:
41
)6)(4()5)(3()1)(2(
BA zzyyxx BABABA
SOLUTION TO EXAMPLE 4
zyx
z
y
x
zyx
zyx
zyx
zyx
BBB
AAA
aaa
a
a
a
aaaaaa
BA
782
)1)(3()5)(2(
)1)(4()6)(2(
)5)(4()6)(3(
651
432
The cross product is:
SOLUTION TO EXAMPLE 4 (Cont’d)
