### Vectors

```Chapter 46
Vectors
ch46 Vectors
by Chtan FYKulai
1
A VECTOR?
□ Describes the motion of an object
□ A Vector comprises
□ Direction
□ Magnitude
Size
□ We will consider
□ Column Vectors
□ General Vectors
□ Vector Geometry
ch46 Vectors
by Chtan FYKulai
2
Column Vectors
NOTE!
Vector a
Label is in BOLD.
2 up
a
When handwritten,
draw a wavy line
under the label
a
i.e. ~
4 RIGHT
 4
 
 2
COLUMN Vector
ch46 Vectors
by Chtan FYKulai
3
Column Vectors
Vector b
2 up
b
3 LEFT
 3 
 2
 
COLUMN Vector?
ch46 Vectors
by Chtan FYKulai
4
Column Vectors
Vector u
2 down
n
4 LEFT
 4 
 2 
 
COLUMN Vector?
ch46 Vectors
by Chtan FYKulai
5
Describe these vectors
4
1
 
a
1
 3
 
b
c
 2 
 3
 
d
ch46 Vectors
by Chtan FYKulai
 4 
 3 
 
6
Alternative labelling
B
D
EF
E
AB
F
CD
G
C
A
GH
H
ch46 Vectors
by Chtan FYKulai
7
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
ch46 Vectors
by Chtan FYKulai
8
General Vectors
Line CD is Parallel to AB
B
A
CD is TWICE length of AB
k
D
2k
Line EF is Parallel to AB
E
C
-k
EF is equal in length to AB
EF is opposite direction to AB
F
ch46 Vectors
by Chtan FYKulai
9
Write these Vectors in terms of k
B
k
D
2k
½k
1½k
F
G
E
A
C
-2k
H
ch46 Vectors
by Chtan FYKulai
10
Combining Column Vectors
 2
k   
1
AB
k
B
AB  3k
D
CD  2k
 2
AB  3  
1
A
 2
CD  2  
1
C
6
AB   
 3
AB
 4
CD   
 2
ch46 Vectors
by Chtan FYKulai
11
Simple combinations
4
AB   
1 
C
1 
BC  
3

 
5
AC =  
4
B
A
a  c   a  c 
      

 b   d  b  d 
ch46 Vectors
by Chtan FYKulai
12
Vector Geometry
Consider this parallelogram
Q
OR  b  PQ
P
R
a
b
O
OP  a  RQ
Opposite sides are Parallel
OQ  OP  PQ
 a +b
OQ  OR  RQ
 b +a
a +b  b + a
OQ is known as the resultant of a and b
ch46 Vectors
by Chtan FYKulai
13
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
14
e.g.1
S is the Midpoint of PQ.
Work out the vector
.
Q
S
P
OS
OS  OP  ½PQ
= a + ½b
R
a
b
O
ch46 Vectors
by Chtan FYKulai
15
Alternatively
S is the Midpoint of PQ.
.
Q
S
P
Work out the vector
OS
OS  OR  RQ  QS
R
a
b
O
= b + a - ½b
= ½b + a
= a + ½b
ch46 Vectors
by Chtan FYKulai
16
e.g.2
C
AC= p, AB = q
p
A
M
q
Find BC
M is the Midpoint of BC
B
BC = BA + AC
= -q + p
=p-q
ch46 Vectors
by Chtan FYKulai
17
e.g.3
C
AC= p, AB = q
p
A
M
q
Find BM
M is the Midpoint of BC
B
BM = ½BC
= ½(p – q)
ch46 Vectors
by Chtan FYKulai
18
e.g.4
C
AC= p, AB = q
p
A
M is the Midpoint of BC
M
q
Find AM
B
AM = AB
+ ½BC
= q + ½(p – q)
= q +½p - ½q
= ½q +½p
ch46 Vectors
= ½(q + p)
by Chtan FYKulai
= ½(p + q)
19
Alternatively
C
AC= p, AB = q
p
A
M is the Midpoint of BC
M
q
Find AM
B
AM = AC + ½CB
= p + ½(q – p)
= p +½q - ½p
= ½p +½q
ch46 Vectors
= ½(p + q)
by Chtan FYKulai
20
Distribution’s law :
The scalar multiplication of a vector :
+  =  +
,
ch46 Vectors
> 0   < 0
by Chtan FYKulai
21
Other important facts :
ℎ  = ℎ
ℎ +   = ℎ +
ch46 Vectors
by Chtan FYKulai
22
A vector with the starting
point from the origin point
is called position vector.

ch46 Vectors
by Chtan FYKulai
23
Every vector can be
expressed in terms of
position vector.
ch46 Vectors
by Chtan FYKulai
24
e.g.5
2
−2
Given that  =
,=
5
3
10
and also  +  =
. Find
1
the values of   .
ch46 Vectors
by Chtan FYKulai
25
e.g.6
Given that  =  − 4,  =
3 − 2, and    are
parallel. Find the value of m.
ch46 Vectors
by Chtan FYKulai
26
e.g.7
2
3
=
,  =
, a point
5
−2
1,4 . Find the coordinates
of   , then express
point  in terms of    .
ch46 Vectors
by Chtan FYKulai
27
e.g.8
5
If  3,5 ,  =
, find the
−7
coordinates of .
ch46 Vectors
by Chtan FYKulai
28
e.g.9
Given that  = 2 + ,  =
7 +   + 4, and
are parallel. Find the value
of .
ch46 Vectors
by Chtan FYKulai
29
Magnitude of a vector
1 , 1 ,   2 , 2 .
=
=
−

+  −
ch46 Vectors
by Chtan FYKulai

30
,

0

=

+

Unit vector :

=
∙

ch46 Vectors
by Chtan FYKulai
31
e.g.10
Find the magnitude of
the vectors :
−
=

(b)  =  −
ch46 Vectors
by Chtan FYKulai
32
e.g.11
Find the unit vectors in
e.g. 10 :
−
=

(b)  =  −
ch46 Vectors
by Chtan FYKulai
33
Ratio theorem

A
P
1
a  b
B p
1 

ch46 Vectors
by Chtan FYKulai
34
e.g.12
M is the midpoint of AB,
find b in terms of a, m .
ch46 Vectors
by Chtan FYKulai
35
e.g.13

2
3

6b
4a
P divides AB
into 2:3. Find
OP in terms
of a, b .

ch46 Vectors
by Chtan FYKulai
36
Application of vector in plane geometry
e.g.14
A
M
X
C
In the diagram, CB=4CN,
NA=5NX, M is the midpoint
of AB.
B
CN  u , BM  v
N
(a) Express the following vectors in terms
of u and v ; (i) NB (ii) NA
ch46 Vectors
by Chtan FYKulai
37
2
(b) Show that CX  4u  v 
5
(c) Calculate the value of
(i) CX
(ii)
Area ACX
CM
Area ACM
ch46 Vectors
by Chtan FYKulai
38
Soln:
(a) (i)
CB  CN  NB
NB  CB  CN  4CN  CN  3CN  3u
(ii)
NA  NB  BA  3u  2v
(b)
1
CX  CN  NX  CN  NA
5
1
8
2
2
 u  3u  2v   u  v  4u  v 
5
5
5
5
ch46 Vectors
by Chtan FYKulai
39
(c) (i) CM  CB  BM  4u  v
2
CX  CM
5
CX
2

CM
5
(ii)
1
CX h 
Area ACX
CX
2
2



1
Area ACM
CM
5
CM h 
2
ch46 Vectors
by Chtan FYKulai
40
e.g.15
M
B
A
M and N are
midpoints of AB, AC.
N Prove that
C
ch46 Vectors
1
MN  BC and MN // BC
2
by Chtan FYKulai
41
e.g.16
A
2a
1
6a
B
1
K
In the diagram K divides AD into
1:l, and divides BC into 1:k .
l
k
O
2b
C
6b
D
Express position vector OK in 2 formats.
Find the values of k and l.
ch46 Vectors
by Chtan FYKulai
42
More exercises on
this topic :

Pg 33 Ex10g
ch46 Vectors
by Chtan FYKulai
43
Scalar product of two vectors
If a and b are two non-zero
vectors, θ is the angle between
the vectors. Then ,
a  b  a b cos 
ch46 Vectors
by Chtan FYKulai
44
Scalar product of vectors satisfying :
Commutative law :
a b  b  a
Associative law :
k a   b  a  k b   k a  b 
Distributive law :
a  b  c   a  b  a  c
ch46 Vectors
by Chtan FYKulai
45
e.g.17
Find the scalar product of the
following 2 vectors :
a  6 , b  5 ,  between is 60
ch46 Vectors
by Chtan FYKulai
46
e.g.18
(a) If a  b  a b , find the angle
between them.
(b) If a  1, b  2, a  k b and a  k b
are perpendicular, find k.
ch46 Vectors
by Chtan FYKulai
47
Scalar product (special cases)
1. Two perpendicular vectors
a  0, b  0,
a  b  a b  0
N.B.
Unit vector for y-axis
i  j  j i  0
Unit vector for x-axis
ch46 Vectors
by Chtan FYKulai
48
2. Two parallel vectors
a  0, b  0,
a // b  a  b   a b
N.B.
i i  1  j  j
 
i   i   1  j   j
ch46 Vectors
by Chtan FYKulai
49
e.g.19
Given a  3, b  8, a  b  2 14 ,
Find
a b
.
Ans:[17/2]
ch46 Vectors
by Chtan FYKulai
50
Scalar product (dot product)
The dot product can also be defined
as the sum of the products of the
components of each vector as :
 x1 
 x2 
a   , b   
 y1 
 y2 
 a  b  x1 x2  y1 y2
ch46 Vectors
by Chtan FYKulai
51
e.g.20
Given that
 3
7
a   ; b   
 4
1
Find (a) a  b
(b) angle between a and b .
Ans: (a) 25 (b) 45°
ch46 Vectors
by Chtan FYKulai
52
Applications of Scalar product

Pg 42 to pg43
Eg30 to eg 33
ch46 Vectors
by Chtan FYKulai
53
More exercises on
this topic :

Pg 44 Ex10i
Misc 10
ch46 Vectors
by Chtan FYKulai
54
Miscellaneous
Examples
ch46 Vectors
by Chtan FYKulai
55
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
AG BG CG


2 .
GD GE GF
ch46 Vectors
by Chtan FYKulai
56
A
Soln:
From ratio theorem
1
d  b  c 
2
1
e  a  c 
2
1
f  a  b 
2
ch46 Vectors
B
by Chtan FYKulai
F
G
D
E
C
57
We select a point G on AD such

that
= .

From ratio theorem,
1
2 1
1
g  a   b  c   a  b  c 
3
3 2
3
Similarly,
We select a G1 point on BE such that

= .

ch46 Vectors
by Chtan FYKulai
58
1
g 1  a  b  c 
3
Similarly,
We select a G2 point on CF such that

= .

1
g 2  a  b  c 
3
ch46 Vectors
by Chtan FYKulai
59
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
=
established.

ch46 Vectors
=

by Chtan FYKulai
=  is
60
Centroid of a ∆
ch46 Vectors
by Chtan FYKulai
61
ch46 Vectors
by Chtan FYKulai
62
The end
ch46 Vectors
by Chtan FYKulai
63
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