review: analysis vector. vector analysis 1.1scalars and vectors 1.2vector components and unit vector...
TRANSCRIPT
![Page 1: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/1.jpg)
Review: Analysis vector
![Page 2: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/2.jpg)
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
![Page 3: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/3.jpg)
• A scalar quantity – has only magnitude• A vector quantity – has both magnitude
and direction
1.1 SCALARS & VECTORS
electric field intensity
![Page 4: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/4.jpg)
• A vector in Cartesian Coordinates maybe represented as
zyx RRR ,,R
1.2 VECTOR COMPONENTS & UNIT VECTOR
R
Or
zzyyxx RRR aaaR
![Page 5: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/5.jpg)
R
yRThe vector has three component vectors, which are , and
zRxR
VECTOR COMPONENTS & UNIT VECTOR (Cont’d)
![Page 6: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/6.jpg)
• 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)
![Page 7: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/7.jpg)
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)
![Page 8: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/8.jpg)
Unit vector in the direction of the vector is:
R
R
RRa
222zyx
RRRR
VECTOR COMPONENTS & UNIT VECTOR (Cont’d)
![Page 9: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/9.jpg)
EXAMPLE 1
Specify the unit vector extending from
the origin toward the point
1,2,2 G
![Page 10: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/10.jpg)
SOLUTION TO EXAMPLE 1
Construct the vector extending from origin to point G
Find the magnitude of
zyx aaaG
22G
3122 222 G
![Page 11: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/11.jpg)
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)
![Page 12: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/12.jpg)
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
![Page 13: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/13.jpg)
VECTOR ALGEBRA (Cont’d)
Vectors in 2 components
![Page 14: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/14.jpg)
• Vector subtraction is similarly carried out as:
)B(ABAD
zzzyyyxxx BABABA aaaD
VECTOR ALGEBRA (Cont’d)
![Page 15: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/15.jpg)
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 )(
![Page 16: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/16.jpg)
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
![Page 17: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/17.jpg)
(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
![Page 18: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/18.jpg)
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)
![Page 19: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/19.jpg)
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)
![Page 20: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/20.jpg)
• 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
![Page 21: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/21.jpg)
zyxP aaar 543
POSITION AND DISTANCE VECTOR (Cont’d)
![Page 22: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/22.jpg)
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
![Page 23: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/23.jpg)
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
![Page 24: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/24.jpg)
(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
![Page 25: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/25.jpg)
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
![Page 26: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/26.jpg)
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
![Page 27: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/27.jpg)
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
![Page 28: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/28.jpg)
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
![Page 29: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/29.jpg)
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)
![Page 30: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/30.jpg)
Properties of cross product of unit vectors:
yxzxzyzyx aaaaaaaaa ,,
Or by using cyclic permutation:
VECTOR PRODUCT OF VECTORS (Cont’d)
![Page 31: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/31.jpg)
Determine the dot product and cross product of the following vectors:
zyx
zyx
aaaB
aaaA
65
432
EXAMPLE 4
![Page 32: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/32.jpg)
The dot product is:
41
)6)(4()5)(3()1)(2(
BA zzyyxx BABABA
SOLUTION TO EXAMPLE 4
![Page 33: Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR](https://reader035.vdocument.in/reader035/viewer/2022081506/56649ef05503460f94c008cb/html5/thumbnails/33.jpg)
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)