vectors
DESCRIPTION
Vectors. Chapter 46. A VECTOR?. Describes the motion of an object A Vector comprises Direction Magnitude We will consider Column Vectors General Vectors Vector Geometry. Size. NOTE! Label is in BOLD . When handwritten, draw a wavy line under the label i.e. a. Column Vectors. - PowerPoint PPT PresentationTRANSCRIPT
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ch46 Vectors by Chtan FYKulai 1
Vectors
Chapter 46
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A VECTOR?□Describes the motion of an object□A Vector comprises
□Direction□Magnitude
□We will consider□Column Vectors□General Vectors□Vector Geometry
Size
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Column Vectors
a
Vector a
COLUMN Vector
4 RIGHT
2 up
NOTE!
Label is in BOLD.
When handwritten, draw a wavy line under the label
i.e. ~a
2
4
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Column Vectors
b
Vector b
COLUMN Vector?
3
2
3 LEFT
2 up
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Column Vectors
n
Vector u
COLUMN Vector?
4
2
4 LEFT
2 down
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Describe these vectors
b
a
c
d
2
3
1
3
4
1
4
3
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Alternative labelling
CD22222222222222
EF22222222222222
AB
A
B
C
DF
E
G
H
GH22222222222222
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General VectorsA Vector has BOTH a Length & a Direction
k can be in any position
k
k
k
k
All 4 Vectors here are EQUAL in Length andTravel in SAME Direction.All called k
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General Vectors
kA
B
C
D
-k
2k
F
E
Line CD is Parallel to AB
CD is TWICE length of AB
Line EF is Parallel to AB
EF is equal in length to AB
EF is opposite direction to AB
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Write these Vectors in terms of k
k
A
B
C
D
E
F G
H
2k1½k ½k
-2k
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Combining Column Vectors
AB
AB
k
A
B
C
D
3k22222222222222AB
1
2k
23
1
22222222222222AB
6
3
22222222222222AB
2k22222222222222CD
22
1
22222222222222CD
4
2
22222222222222CD
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A
B
C
Simple combinations
1
4AB
5AC =
4
22222222222222
3
1BC
db
ca
d
c
b
a
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Vector Geometry
OP a22222222222222
OR b22222222222222
RQ22222222222222Consider this parallelogram
Q
O
P
Ra
b
PQ22222222222222
Opposite sides are Parallel
OQ OP PQ222222222222222222222222222222222222222222
OQ OR RQ222222222222222222222222222222222222222222
OQ is known as the resultant of a and b
a+b
b + a
a+b b + a
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Resultant of Two Vectors
□Is the same, no matter which route is followed
□Use this to find vectors in geometrical figures
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e.g.1
Q
O
P
Ra
b
.S
S is the Midpoint of PQ.
Work out the vector OS
PQOPOS ½
= a + ½b
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Alternatively
Q
O
P
Ra
b
.SS is the Midpoint of PQ.
Work out the vector OS
OS OR RQ QS 22222222222222222222222222222222222222222222222222222222
= a + ½b
= b + a - ½b
= ½b + a
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AB
C
p
q
M M is the Midpoint of BC
Find BC
AC= p, AB = q
BC BA AC= += -q + p
= p - q
e.g.2
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AB
C
p
q
M M is the Midpoint of BC
Find BM
AC= p, AB = q
BM ½BC=
= ½(p – q)
e.g.3
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AB
C
p
q
M M is the Midpoint of BC
Find AM
AC= p, AB = q
= q + ½(p – q)
AM + ½BC= AB
= q +½p - ½q
= ½q +½p = ½(q + p) = ½(p + q)
e.g.4
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Alternatively
AB
C
p
q
M M is the Midpoint of BC
Find AM
AC= p, AB = q
= p + ½(q – p)
AM + ½CB= AC
= p +½q - ½p
= ½p +½q = ½(p + q)
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Distribution’s law :
𝑘 (𝒂+𝒃 )=𝑘𝒂+𝑘𝒃
The scalar multiplication of a vector :
𝑘𝑖𝑠 𝑎𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ,𝑘>0𝑜𝑟𝑘<0
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Other important facts :
h𝑘 (𝒂 )=( h𝑘) 𝒂
(h+𝑘 )𝒂=h𝒂+𝑘𝒂
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A vector with the starting point from the origin point is called position vector.
位置向量
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Every vector can be expressed in terms of position vector.
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e.g.5
Given that , and also Find the values of
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e.g.6
Given that ,, and are parallel. Find the value of
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e.g.7
=, , a point . Find the coordinates of then express point in terms of .
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e.g.8
If , , find the coordinates of
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e.g.9
Given that ,, and are parallel. Find the value of
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Magnitude of a vector
𝐴𝑖𝑠 (𝑥1 , 𝑦1 ) ,𝐵𝑖𝑠 (𝑥2 , 𝑦 2 ) .
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(𝒙 ,𝒚 )𝒂
0
𝑦
𝑥
|𝒂|=√𝒙𝟐+𝒚𝟐
Unit vector :
�̂�=𝟏|𝒂|
∙𝒂
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e.g.10
Find the magnitude of the vectors :
(b)
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e.g.11
Find the unit vectors in e.g. 10 :
(b)
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ch46 Vectors by Chtan FYKulai 34
Ratio theorem
𝒙
𝒚
𝟎
P A
B
1
1
bap
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e.g.12
M is the midpoint of AB, find in terms of .
b ma,
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e.g.13
𝑨 𝑩
𝑶
𝑷
a4 b6
2 3 P divides AB into 2:3. Find in terms of .
OPba,
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ch46 Vectors by Chtan FYKulai 37
Application of vector in plane geometry
e.g.14A
B
C
M
N
X
In the diagram, CB=4CN, NA=5NX, M is the midpoint of AB.
vBMuCN ,
(a) Express the following vectors in terms of u and v ; (i) (ii)NB NA
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(b) Show that vuCX 45
2
(c) Calculate the value of (i) (ii)
CM
CX
ACMArea
ACXArea
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Soln:(a) (i) NBCNCB
uCNCNCNCNCBNB 334
(ii) vuBANBNA 23
(b) NACNNXCNCX5
1
vuvuvuu 45
2
5
2
5
823
5
1
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(c) (i)
5
25
2
4
CM
CX
CMCX
vuBMCBCM
(ii)
5
2
2121
CM
CX
hCM
hCX
ACMArea
ACXArea
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e.g.15 A
BC
M N
M and N are midpoints of AB, AC.Prove that
BCMNBCMN // and 2
1
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e.g.16
A
B
CD
K
O
l6a
1
12a
k
2b 6b
In the diagram K divides AD into 1:l, and divides BC into 1:k .
Express position vector OK in 2 formats. Find the values of k and l.
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ch46 Vectors by Chtan FYKulai 43
高级数学高二下册Pg 33 Ex10g
More exercises on this topic :
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Scalar product of two vectors
If a and b are two non-zero vectors, θ is the angle between the vectors. Then ,
cosbaba
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Scalar product of vectors satisfying :
Commutative law : abba
Associative law :
bakbkabak
Distributive law :
cabacba
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e.g.17
Find the scalar product of the following 2 vectors :
60 isbetween , 5 , 6 ba
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e.g.18
(a)If , find the angle between them.
(b)If
are perpendicular, find k.
baba
,2,1 ba bkabka and
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Scalar product (special cases)
1. Two perpendicular vectors
0
,0,0
baba
ba
N.B.0 ijji
Unit vector for x-axis
Unit vector for y-axis
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2. Two parallel vectors
bababa
ba
//
,0,0
N.B.
jjii
jjii
1
1
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e.g.19
Given ,
Find .
142,8,3 baba
ba
Ans:[17/2]
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Scalar product (special cases)
The dot product can also be defined as the sum of the products of the components of each vector as :
2
2
1
1 ,y
xb
y
xa
2121 yyxxba
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e.g.20
Given that
1
7;
4
3ba
Find (a) (b) angle between a and b .
ba
Ans: (a) 25 (b) 45°
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Applications of Scalar product
高级数学高二下册Pg 42 to pg43Eg30 to eg 33
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ch46 Vectors by Chtan FYKulai 54
高级数学高二下册Pg 44 Ex10iMisc 10
More exercises on this topic :
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The end