vectors chapter 46 ch46 vectors by chtan fykulai 1

ch46 Vectors by Chtan FYKulai 1

Vectors

Chapter 46

ch46 Vectors by Chtan FYKulai 2

A VECTOR?□Describes the motion of an object□A Vector comprises

□Direction□Magnitude

□We will consider□Column Vectors□General Vectors□Vector Geometry

Size

ch46 Vectors by Chtan FYKulai 3

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

ch46 Vectors by Chtan FYKulai 4

Column Vectors

b

Vector b

COLUMN Vector?

3

2

3 LEFT

2 up

ch46 Vectors by Chtan FYKulai 5

Column Vectors

n

Vector u

COLUMN Vector?

4

2

4 LEFT

2 down

ch46 Vectors by Chtan FYKulai 6

Describe these vectors

b

a

c

d

2

3

1

3

4

1

4

3

ch46 Vectors by Chtan FYKulai 7

Alternative labelling

CD22222222222222

EF22222222222222

AB

A

B

C

DF

E

G

H

GH22222222222222

ch46 Vectors by Chtan FYKulai 8

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

ch46 Vectors by Chtan FYKulai 9

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

ch46 Vectors by Chtan FYKulai 10

Write these Vectors in terms of k

k

A

B

C

D

E

F G

H

2k1½k ½k

-2k

ch46 Vectors by Chtan FYKulai 11

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

ch46 Vectors by Chtan FYKulai 12

A

B

C

Simple combinations

1

4AB

5AC =

4

22222222222222

3

1BC

db

ca

d

c

b

a

ch46 Vectors by Chtan FYKulai 13

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

ch46 Vectors by Chtan FYKulai 14

Resultant of Two Vectors

□Is the same, no matter which route is followed

□Use this to find vectors in geometrical figures

ch46 Vectors by Chtan FYKulai 15

e.g.1

Q

O

P

Ra

b

.S

S is the Midpoint of PQ.

Work out the vector OS

PQOPOS ½

= a + ½b

ch46 Vectors by Chtan FYKulai 16

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

ch46 Vectors by Chtan FYKulai 17

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

ch46 Vectors by Chtan FYKulai 18

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

ch46 Vectors by Chtan FYKulai 19

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

ch46 Vectors by Chtan FYKulai 20

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)

ch46 Vectors by Chtan FYKulai 21

Distribution’s law :

𝑘 (𝒂+𝒃 )=𝑘𝒂+𝑘𝒃

The scalar multiplication of a vector :

𝑘𝑖𝑠 𝑎𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ,𝑘>0𝑜𝑟𝑘<0

ch46 Vectors by Chtan FYKulai 22

Other important facts :

h𝑘 (𝒂 )=( h𝑘) 𝒂

(h+𝑘 )𝒂=h𝒂+𝑘𝒂

ch46 Vectors by Chtan FYKulai 23

A vector with the starting point from the origin point is called position vector.

位置向量

ch46 Vectors by Chtan FYKulai 24

Every vector can be expressed in terms of position vector.

ch46 Vectors by Chtan FYKulai 25

e.g.5

Given that , and also Find the values of

ch46 Vectors by Chtan FYKulai 26

e.g.6

Given that ,, and are parallel. Find the value of

ch46 Vectors by Chtan FYKulai 27

e.g.7

=, , a point . Find the coordinates of then express point in terms of .

ch46 Vectors by Chtan FYKulai 28

e.g.8

If , , find the coordinates of

ch46 Vectors by Chtan FYKulai 29

e.g.9

Given that ,, and are parallel. Find the value of

ch46 Vectors by Chtan FYKulai 30

Magnitude of a vector

𝐴𝑖𝑠 (𝑥1 , 𝑦1 ) ,𝐵𝑖𝑠 (𝑥2 , 𝑦 2 ) .

ch46 Vectors by Chtan FYKulai 31

(𝒙 ,𝒚 )𝒂

0

𝑦

𝑥

|𝒂|=√𝒙𝟐+𝒚𝟐

Unit vector :

�̂�=𝟏|𝒂|

∙𝒂

ch46 Vectors by Chtan FYKulai 32

e.g.10

Find the magnitude of the vectors :

(b)

ch46 Vectors by Chtan FYKulai 33

e.g.11

Find the unit vectors in e.g. 10 :

(b)

ch46 Vectors by Chtan FYKulai 34

Ratio theorem

𝒙

𝒚

𝟎

P A

B

1

1

bap

ch46 Vectors by Chtan FYKulai 35

e.g.12

M is the midpoint of AB, find in terms of .

b ma,

ch46 Vectors by Chtan FYKulai 36

e.g.13

𝑨 𝑩

𝑶

𝑷

a4 b6

2 3 P divides AB into 2:3. Find in terms of .

OPba,

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

ch46 Vectors by Chtan FYKulai 38

(b) Show that vuCX 45

2

(c) Calculate the value of (i) (ii)

CM

CX

ACMArea

ACXArea

ch46 Vectors by Chtan FYKulai 39

Soln:(a) (i) NBCNCB

uCNCNCNCNCBNB 334

(ii) vuBANBNA 23

(b) NACNNXCNCX5

1

vuvuvuu 45

2

5

2

5

823

5

1

ch46 Vectors by Chtan FYKulai 40

(c) (i)

5

25

2

4

CM

CX

CMCX

vuBMCBCM

(ii)

5

2

2121

CM

CX

hCM

hCX

ACMArea

ACXArea

ch46 Vectors by Chtan FYKulai 41

e.g.15 A

BC

M N

M and N are midpoints of AB, AC.Prove that

BCMNBCMN // and 2

1

ch46 Vectors by Chtan FYKulai 42

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.

ch46 Vectors by Chtan FYKulai 43

高级数学高二下册Pg 33 Ex10g

More exercises on this topic :

ch46 Vectors by Chtan FYKulai 44

Scalar product of two vectors

If a and b are two non-zero vectors, θ is the angle between the vectors. Then ,

cosbaba

ch46 Vectors by Chtan FYKulai 45

Scalar product of vectors satisfying :

Commutative law : abba

Associative law :

bakbkabak

Distributive law :

cabacba

ch46 Vectors by Chtan FYKulai 46

e.g.17

Find the scalar product of the following 2 vectors :

60 isbetween , 5 , 6 ba

ch46 Vectors by Chtan FYKulai 47

e.g.18

(a)If , find the angle between them.

(b)If

are perpendicular, find k.

baba

,2,1 ba bkabka and

ch46 Vectors by Chtan FYKulai 48

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

ch46 Vectors by Chtan FYKulai 49

2. Two parallel vectors

bababa

ba

//

,0,0

N.B.

jjii

jjii

1

1

ch46 Vectors by Chtan FYKulai 50

e.g.19

Given ,

Find .

142,8,3 baba

ba

Ans:[17/2]

ch46 Vectors by Chtan FYKulai 51

Scalar product (dot product)

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

ch46 Vectors by Chtan FYKulai 52

e.g.20

Given that

1

7;

4

3ba

Find (a) (b) angle between a and b .

ba

Ans: (a) 25 (b) 45°

ch46 Vectors by Chtan FYKulai 53

Applications of Scalar product

高级数学高二下册Pg 42 to pg43Eg30 to eg 33

ch46 Vectors by Chtan FYKulai 54

高级数学高二下册Pg 44 Ex10iMisc 10

More exercises on this topic :

ch46 Vectors by Chtan FYKulai 55

Miscellaneous Examples

ch46 Vectors by Chtan FYKulai 56

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

.2GF

CG

GE

BG

GD

AG

ch46 Vectors by Chtan FYKulai 57

Soln: A

B CD

EFG

From ratio theorem

cbd 2

1

cae 2

1

baf 2

1

ch46 Vectors by Chtan FYKulai 58

We select a point G on AD such that

From ratio theorem,

cbacbag 3

1

2

1

3

2

3

1

Similarly,We select a G1 point on BE such that

ch46 Vectors by Chtan FYKulai 59

cbag 3

11

Similarly,

We select a G2 point on CF such that

cbag 3

12

ch46 Vectors by Chtan FYKulai 60

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 is established.

ch46 Vectors by Chtan FYKulai 61

Centroid of a ∆

ch46 Vectors by Chtan FYKulai 62

ch46 Vectors by Chtan FYKulai 63

The end