1. ch46 Vectors by Chtan FYKulai
Chapter 46
Vectors
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2. ch46 Vectors by Chtan FYKulai
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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3. ch46 Vectors by Chtan FYKulai
Column Vectors
NOTE!
Label is in BOLD.
When handwritten,
draw a wavy line
under the label
i.e.
Vector a a
2 up
4 RIGHT
COLUMN Vector
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4. ch46 Vectors by Chtan FYKulai
Column Vectors
Vector b b
2 up
3 LEFT
COLUMN Vector?
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5. ch46 Vectors by Chtan FYKulai
Column Vectors
Vector u n
2 down
4 LEFT
COLUMN Vector?
ch46 Vectors by Chtan FYKulai

6. Describe these vectorsa b c d
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7. Alternative labellingF B D E G C A H
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8. ch46 Vectors by Chtan FYKulai
General Vectors
A Vector has BOTH a Length & a Direction
All 4 Vectors here are EQUAL in Length and
Travel in SAME Direction.
All called k k k k k
k can be in any position
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9. ch46 Vectors by Chtan FYKulai
General Vectors B
Line CD is Parallel to AB k
CD is TWICE length of AB D A 2k
Line EF is Parallel to AB E C
EF is equal in length to AB -k
EF is opposite direction to AB F
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10. Write these Vectors in terms of kB D 2k F G
1½k ½k E C
-2k A H
ch46 Vectors by Chtan FYKulai

11. Combining Column Vectorsk A B C D
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12. ch46 Vectors by Chtan FYKulai
Simple combinations A B C
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13. ch46 Vectors by Chtan FYKulai
Vector Geometry
Consider this parallelogram Q O P R a b
Opposite sides are Parallel
OQ is known as the resultant of a and b
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14. 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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15. ch46 Vectors by Chtan FYKulai
e.g.1
S is the Midpoint of PQ.
Work out the vector Q O P R a b . S
= a + ½b
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16. ch46 Vectors by Chtan FYKulai
Alternatively
S is the Midpoint of PQ.
Work out the vector Q O P R a b . S
= b
+ a
- ½b
= ½b + a
= a + ½b
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17. ch46 Vectors by Chtan FYKulai
e.g.2 A B C p q M
AC= p, AB = q
M is the Midpoint of BC
Find BC BC BA AC = +
= -q + p
= p - q
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18. ch46 Vectors by Chtan FYKulai
e.g.3 A B C p q M
AC= p, AB = q
M is the Midpoint of BC
Find BM BM
½BC =
= ½(p – q)
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19. ch46 Vectors by Chtan FYKulai
e.g.4 A B C p q M
AC= p, AB = q
M is the Midpoint of BC
Find AM AM
+ ½BC = AB
= q + ½(p – q)
= q +½p - ½q
= ½q +½p
= ½(q + p)
= ½(p + q)
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20. ch46 Vectors by Chtan FYKulai
Alternatively A B C p q M
AC= p, AB = q
M is the Midpoint of BC
Find AM AM
+ ½CB = AC
= p + ½(q – p)
= p +½q - ½p
= ½p +½q
= ½(p + q)
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21. ch46 Vectors by Chtan FYKulai
Distribution’s law :
The scalar multiplication of a vector :
𝑘 𝒂+𝒃 =𝑘𝒂+𝑘𝒃
𝑘 𝑖𝑠 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, 𝑘>0 𝑜𝑟 𝑘<0
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22. ch46 Vectors by Chtan FYKulai
Other important facts :
ℎ𝑘 𝒂 = ℎ𝑘 𝒂
ℎ+𝑘 𝒂=ℎ𝒂+𝑘𝒂
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23. ch46 Vectors by Chtan FYKulai
A vector with the starting point from the origin point is called position vector.
位置向量
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24. ch46 Vectors by Chtan FYKulai
Every vector can be expressed in terms of position vector.
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25. ch46 Vectors by Chtan FYKulai
e.g.5
Given that 𝒂= , 𝒃= − and also 𝑘𝒂+𝑙𝒃= Find the values of 𝑘 𝑎𝑛𝑑 𝑙.
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26. ch46 Vectors by Chtan FYKulai
e.g.6
Given that 𝒂=𝑚𝑖−4𝑗, 𝒃=3𝑖−2𝑗, and 𝒂 𝒂𝒏𝒅 𝒃 are parallel. Find the value of m.
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27. ch46 Vectors by Chtan FYKulai
e.g.7
𝐴𝐵 = 3 −2 , 𝐵𝐶 = , a point 𝐶 1,4 . Find the coordinates of 𝐴 𝑎𝑛𝑑 𝐵, then express point 𝐶 in terms of 𝒊 𝑎𝑛𝑑 𝒋 .
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28. ch46 Vectors by Chtan FYKulai
e.g.8
If 𝑃 3,5 , 𝑃𝑄 = 5 −7 , find the coordinates of 𝑄.
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29. ch46 Vectors by Chtan FYKulai
e.g.9
Given that 𝒂=2𝑖+𝑝𝑗, 𝒃= 7+𝑝 𝑖+4𝑗, and 𝒂 𝒂𝒏𝒅 𝒃 are parallel. Find the value of 𝑝.
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30. ch46 Vectors by Chtan FYKulai
Magnitude of a vector
𝐴 𝑖𝑠 𝑥 1 , 𝑦 1 , 𝐵 𝑖𝑠 𝑥 2 , 𝑦 2 .
𝒂 =𝑨𝑩
= 𝒙 𝟐 − 𝒙 𝟏 𝟐 + 𝒚 𝟐 − 𝒚 𝟏 𝟐
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31. ch46 Vectors by Chtan FYKulai
𝒙,𝒚 𝒂
𝒂 = 𝒙 𝟐 + 𝒚 𝟐 𝑦 𝑥
Unit vector :
𝒂 = 𝟏 𝒂 ∙𝒂
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32. ch46 Vectors by Chtan FYKulai
e.g.10
Find the magnitude of the vectors :
𝒂 𝒑= −𝟐 𝟓
(b) 𝒓=𝟗𝒊−𝟏𝟐𝒋
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33. ch46 Vectors by Chtan FYKulai
e.g.11
Find the unit vectors in e.g. 10 :
𝒂 𝒑= −𝟐 𝟓
(b) 𝒓=𝟗𝒊−𝟏𝟐𝒋
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34. ch46 Vectors by Chtan FYKulai
Ratio theorem 𝒚 A P B 𝒙 𝟎
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35. ch46 Vectors by Chtan FYKulai
e.g.12
M is the midpoint of AB, find in terms of
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36. ch46 Vectors by Chtan FYKulai
e.g.13 2 𝑷 3
P divides AB into 2:3. Find
in terms of 𝑨 𝑩 𝑶
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37. ch46 Vectors by Chtan FYKulai
Application of vector in plane geometry
e.g.14
In the diagram, CB=4CN, NA=5NX, M is the midpoint of AB. A M X B C N
(a) Express the following vectors in terms of u and v ; (i) (ii)
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38. ch46 Vectors by Chtan FYKulai
(b) Show that
(c) Calculate the value of
(i) (ii)
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39. ch46 Vectors by Chtan FYKulai
Soln:
(a) (i)
(ii)
(b)
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40. ch46 Vectors by Chtan FYKulai
(c) (i)
(ii)
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41. ch46 Vectors by Chtan FYKulai
e.g.15 A
M and N are midpoints of AB, AC.
Prove that N M C B
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42. ch46 Vectors by Chtan FYKulai
e.g.16 B
In the diagram K divides AD into 1:l, and divides BC into 1:k . 2a 1 A 1 K 6a l k D O C 2b 6b
Express position vector OK in 2 formats. Find the values of k and l.
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43. ch46 Vectors by Chtan FYKulai
More exercises on this topic :
高级数学高二下册
Pg 33 Ex10g
ch46 Vectors by Chtan FYKulai

44. ch46 Vectors by Chtan FYKulai
Scalar product of two vectors
If a and b are two non-zero vectors, θ is the angle between the vectors. Then ,
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45. ch46 Vectors by Chtan FYKulai
Scalar product of vectors satisfying :
Commutative law :
Associative law :
Distributive law :
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46. ch46 Vectors by Chtan FYKulai
e.g.17
Find the scalar product of the following 2 vectors :
ch46 Vectors by Chtan FYKulai

47. ch46 Vectors by Chtan FYKulai
e.g.18
If , find the angle between them. If
are perpendicular, find k.
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48. ch46 Vectors by Chtan FYKulai
Scalar product (special cases)
1. Two perpendicular vectors
Unit vector for y-axis
N.B.
Unit vector for x-axis
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49. ch46 Vectors by Chtan FYKulai
2. Two parallel vectors
N.B.
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50. ch46 Vectors by Chtan FYKulai
e.g.19
Given ,
Find
Ans:[17/2]
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51. ch46 Vectors by Chtan FYKulai
Scalar product (dot product)
The dot product can also be defined as the sum of the products of the components of each vector as :
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52. ch46 Vectors by Chtan FYKulai
e.g.20
Given that
Find (a)
(b) angle between a and b .
Ans: (a) 25 (b) 45°
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53. ch46 Vectors by Chtan FYKulai
Applications of Scalar product
高级数学高二下册
Pg 42 to pg43
Eg30 to eg 33
ch46 Vectors by Chtan FYKulai

54. ch46 Vectors by Chtan FYKulai
More exercises on this topic :
高级数学高二下册
Pg 44 Ex10i
Misc 10
ch46 Vectors by Chtan FYKulai

55. Miscellaneous Examples
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56. ch46 Vectors by Chtan FYKulai
e.g.21
Given that D, E, F are three midpoints of BC, CA, AB of a triangle ABC. Prove that AD, BE and CF are concurrent at a point G and .
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57. ch46 Vectors by Chtan FYKulai
Soln: A F E G
From ratio theorem B D C
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58. ch46 Vectors by Chtan FYKulai
We select a point G on AD such that 𝑨𝑮 𝑮𝑫 =𝟐.
From ratio theorem,
Similarly,
We select a G1 point on BE such that 𝑩 𝑮 𝟏 𝑮 𝟏 𝑬 =𝟐.
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59. ch46 Vectors by Chtan FYKulai
Similarly,
We select a G2 point on CF such that 𝑪 𝑮 𝟐 𝑮 𝟐 𝑭 =𝟐.
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60. ch46 Vectors by Chtan FYKulai
Because g1, g2, g are the same,
G, G1, G2 are the same point G!
G is on AD, BE and CF, hence AD, BE and CF intersect at G.
And also 𝑨𝑮 𝑮𝑫 = 𝑩𝑮 𝑮𝑬 = 𝑪𝑮 𝑮𝑭 =𝟐 is established.
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61. ch46 Vectors by Chtan FYKulai
Centroid of a ∆
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62. ch46 Vectors by Chtan FYKulai

63. ch46 Vectors by Chtan FYKulai
The end
ch46 Vectors by Chtan FYKulai