Vectors Chapter 46 ch46 Vectors by Chtan FYKulai 1.

ch46 Vectors by Chtan FYKulai 1

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


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


Column Vectors

b

Vector b

COLUMN Vector?

3

2

3 LEFT

2 up


Column Vectors

n

Vector u

COLUMN Vector?

4

2

4 LEFT

2 down


Describe these vectors

b

a

c

d

2

3

1

3

4

1

4

3


Alternative labelling

CD22222222222222

EF22222222222222

AB

A

B

C

DF

E

G

H

GH22222222222222


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


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


Write these Vectors in terms of k

k

A

B

C

D

E

F G

H

2k1½k ½k

-2k


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


A

B

C

Simple combinations

1

4AB

5AC =

4

22222222222222

3

1BC

db

ca

d

c

b

a


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


Resultant of Two Vectors

□Is the same, no matter which route is followed

□Use this to find vectors in geometrical figures


e.g.1

Q

O

P

Ra

b

.S

S is the Midpoint of PQ.

Work out the vector OS

PQOPOS ½

= a + ½b


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


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


AB

C

p

q


Find BM

AC= p, AB = q

BM ½BC=

= ½(p – q)

e.g.3


AB

C

p

q


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


Alternatively

AB

C

p

q


Find AM

AC= p, AB = q

= p + ½(q – p)

AM + ½CB= AC

= p +½q - ½p

= ½p +½q = ½(p + q)


Distribution’s law :

𝑘 (𝒂+𝒃 )=𝑘𝒂+𝑘𝒃

The scalar multiplication of a vector :

𝑘𝑖𝑠 𝑎𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ,𝑘>0𝑜𝑟𝑘<0


Other important facts :

h𝑘 (𝒂 )=( h𝑘) 𝒂

(h+𝑘 )𝒂=h𝒂+𝑘𝒂


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

位置向量


Every vector can be expressed in terms of position vector.


e.g.5

Given that , and also Find the values of


e.g.6

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


e.g.7

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


e.g.8

If , , find the coordinates of


e.g.9

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


Magnitude of a vector

𝐴𝑖𝑠 (𝑥1 , 𝑦1 ) ,𝐵𝑖𝑠 (𝑥2 , 𝑦 2 ) .


(𝒙 ,𝒚 )𝒂

0

𝑦

𝑥

|𝒂|=√𝒙𝟐+𝒚𝟐

Unit vector :

�̂�=𝟏|𝒂|

∙𝒂


e.g.10

Find the magnitude of the vectors :

(b)


e.g.11

Find the unit vectors in e.g. 10 :

(b)


Ratio theorem

𝒙

𝒚

𝟎

P A

B

1

1

bap


e.g.12

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

b ma,


e.g.13

𝑨 𝑩

𝑶

𝑷

a4 b6

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

OPba,


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


(b) Show that vuCX 45

2

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

CM

CX

ACMArea

ACXArea


Soln:(a) (i) NBCNCB

uCNCNCNCNCBNB 334

(ii) vuBANBNA 23

(b) NACNNXCNCX5

1

vuvuvuu 45

2

5

2

5

823

5

1


(c) (i)

5

25

2

4

CM

CX

CMCX

vuBMCBCM

(ii)

5

2

2121

CM

CX

hCM

hCX

ACMArea

ACXArea


e.g.15 A

BC

M N

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

BCMNBCMN // and 2

1


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.


高级数学高二下册Pg 33 Ex10g

More exercises on this topic :


Scalar product of two vectors

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

cosbaba


Scalar product of vectors satisfying :

Commutative law : abba

Associative law :

bakbkabak

Distributive law :

cabacba


e.g.17

Find the scalar product of the following 2 vectors :

60 isbetween , 5 , 6 ba


e.g.18

(a)If , find the angle between them.

(b)If

are perpendicular, find k.

baba

,2,1 ba bkabka and


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


2. Two parallel vectors

bababa

ba

//

,0,0

N.B.

jjii

jjii

1

1


e.g.19

Given ,

Find .

142,8,3 baba

ba

Ans:[17/2]


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


e.g.20

Given that

1

7;

4

3ba

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

ba

Ans: (a) 25 (b) 45°


Applications of Scalar product

高级数学高二下册Pg 42 to pg43Eg30 to eg 33


高级数学高二下册Pg 44 Ex10iMisc 10

More exercises on this topic :


Miscellaneous Examples


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


Soln: A

B CD

EFG

From ratio theorem

cbd 2

1

cae 2

1

baf 2

1


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


cbag 3

11

Similarly,

We select a G2 point on CF such that

cbag 3

12


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.


Centroid of a ∆



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Vectors Chapter 46 ch46 Vectors by Chtan FYKulai 1.

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