CHAPTER 3.pptx

CHAPTER 3:
VECTORS
Dr Yusmye Nur Abu Yusuf

INTRODUCTION
Definition 3.1
 VECTOR is a mathematical quantity that has both MAGNITUDE
AND DIRECTION
 VECTOR: represented by arrow where the direction of arrow
indicates the DIRECTION of the vector & the length of arrow
indicates the MAGNITUDE of the vector.
 Eg: displacement, velocity, acceleration, force, ect

INTRODUCTION
Definition 3.2
 SCALAR is a mathematical quantity that has MAGNITUDE
only.
 VECTOR: represented by a single letter s.a, a.
 Eg: temperature, mass, length area, ect

INTRODUCTION
Definition 3.3
 A vector in the plane is a directed line segment that has initial
point A and terminal point B, denoted by, ; its length is
denoted by or also can be represent as .
AB
AB
Initial Point, A
Terminal Point, B
AB
Magnitude (Length): AB
Direction :
u

INTRODUCTION
Definition 3.4
 Two vectors, and are said to be EQUAL if and only if they
have the same magnitude and direction.
u
u
b
a
b
a 


v
v
P
Q
P
Q

Negative Vectors
• Vector having same length as a particular vector but opposite
direction are called as negative vectors.
• Geometrically if then
NHAA/IMK/UNIMAP
u

Q
P
u
Q
P
PQ u
 QP u
 

Parallel Vector
• Given 2 vector are parallel, then any vector and can be
written as .
• There are 2 ways vector can be parallel either they have same
direction or opposite direction. The length doesn’t matter.
Example :
If vector is parallel to , find the value of t if this
two vector are parallel.
u v
u kv

2, 1
u   ,2
v t

2, 1 ,2
2, 1 ,2
2 1 , 2
1 1
, 2, 4
2 2
u kv
k t
tk k
then
k tk
k t t

 
 
  
 
     
 
 

Determine either the given vector are parallel to or not
a)
b)
c)
3, 2,5
v   
6,4, 10
w  
4 10
2, ,
3 3
m  
3 5
, 1,
2 2
z   

COMPONENTS OF VECTORS IN
TERM OF UNIT VACTORS
If is any vector with the initial point at the origin , then
the terminal point of has coordinate . This coordinate
is called as the components of and we write as
Where are unit vectors in the x, y and z axis direction
respectively.
Unit Vectors
u  
0,0,0
O
 
, ,
Q x y z
u
u
or , ,
u xi yj zk x y z
  
, and
i j k
1,0,0 be a unit vector in the OX direction (x-axis)
i 
0,1,0 be a unit vector in the OY direction (y-axis)
j 
0,0,1 be a unit vector in the OZ direction (z-axis)
z 

VECTOR IN COMPONENT FORM
Definition :Position vector
The position vector of the point Q with the coordinates (x,y) is
Where represents the vector from the origin to the terminal
point Q in (2 Dimensional)
Definition : Component Form
 If is a vector with initial point and terminal
point , then the standard COMPONENT FORM
is
   
, 0,0 ,
u OQ x y x y
   
u
2
R
PQ
1 2 3 1 2 3 1 1 2 2 3 3
, , , , , ,
PQ OQ OP q q q p p p q p q p q p
       
 
1 2 3
, ,
P p p p
 
1 2 3
, ,
Q q q q

Definition : Magnitude / Length
 the magnitude of the vector is
Example :
Find (a) component form and (b) length of the vector with
initial point P(-3,4,1) and terminal point Q(-5,2,2)
P
Q
v

     
2 2 2
2 1 2 1 2 1
PQ q p q p q p
     
2 2 2
1 2 3
v v v v
  

Exercise :
Given P(2,1,5) , Q(-2,0,5) and R(4,2,7) are 3 points in , find :
a)
b)
c)
d)
3
R
PQ
RP
PR
QR

UNIT VECTORS
Definition :
if u is a vector, then the unit vector in the direction of u is
defined as:
A vector which have magnitude (length) equal to 1 is called a
unit vector.
u
u
u 


Exercise :
 Given and Find and the unit
vector in the direction of .
 Find the unit vector that has the same direction as
 Find the unit vector in the direction opposite to the direction of
the vectors
a)
b)
 Find the unit vector that points in the direction of the vector
from P(-1,2,5) and Q(0,-3,7). Prove the magnitude is 1.
4
u i j
  2 6 .
v i j
  2 5
w u v
 
w w
2 2
v i j k
  
5 7
u i k
 
2 5 6
w i j k
   
PQ

VECTOR ALGEBRA
OPERATIONS
Definition : Vector Addition and Multiplication by a Scalar
Let and be vectors with k a scalar.
ADDITION :
SCALAR MULTIPLICATION:
3
2
1 ,
, u
u
u
u  3
2
1 ,
, v
v
v
v 
3
3
2
2
1
1 ,
, v
u
v
u
v
u
v
u 




3
2
1 ,
, ku
ku
ku
u
k 

ADDITION OF
VECTORS
The Triangle Law
2 vectors u and v represented by the line segment can be added
by joining the initial point of vector v to the terminal point of u.
The Parallelogram Law
The sum, called the resultant vector is the diagonal of the
parallelogram.

VECTOR ADDITION
u
v
m
m u v
 
v
u
w
w u v
 

u v
x
m
n
w
y
This addition of vectors is satisfied the commutative law  
u v v u
  

SCALAR MULTIPLICATIONS OF
VECTORS
Definition :
Let k be a scalar and u represent a vector, the scalar
multiplication ku is:
i) A vector whose length |k| time of the length u and
ii) A vector whose direction is:
i) The same as u if k>0 and
ii) The opposite direction from u if k<0

• Let be a scalar and be non-zero vector, then has the
magnitude
u
u ku
k
2u
k u

SUBTRACTION OF
VECTORS
The subtraction of 2 vectors, u and v is defined by:
If and then,
 
v
u
v
u 



3
2
1 ,
, u
u
u
u  3
2
1 ,
, v
v
v
v 
3
3
2
2
1
1 ,
, v
u
v
u
v
u
v
u 





Example :
Let and . Find:
1
,
3
,
1


u 0
,
7
,
4

v
u
c
v
u
b
v
u
a
2
1
)
(
3
)
(
3
2
)
(



PROPERTIES OF VECTOR
OPERATIONS
Let u,v,w be vectors and a,b be scalars:
   
 
0
0
)
5
0
)
4
0
)
3
)
2
)
1














u
u
u
u
u
w
v
u
w
v
u
u
v
v
u
   
 
  u
b
u
a
u
b
a
v
a
u
a
v
u
a
u
ab
u
b
a
u
u








)
9
)
8
)
7
1
)
6

DOT PRODUCT
Also known as inner product or scalar product
The result is a scalar
If and then:
3
2
1 ,
, u
u
u
u  3
2
1 ,
, v
v
v
v 
3
3
2
2
1
1
3
2
1
3
2
1 ,
,
.
,
,
.
v
u
v
u
v
u
v
v
v
u
u
u
v
u





Example 2
If and :
 
 
 
.
2 .
( ) 3u 4
i u v
ii u v
iii u v

 
1
,
3
,
2 

u 3
,
2
,
0 

v

PROPERTIES OF DOT
PRODUCT
 
   
    
       
 
 
 
  2
.
0
.
.
.
0
.
1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
u
u
u
vi
i
k
k
j
j
i
vii
v
u
vi
i
i
a
a
u
u
v
v
k
u
v
u
k
v
u
k
iv
p
v
w
v
p
u
w
u
p
w
v
u
iii
w
u
v
u
w
v
u
ii
u
v
v
u
i




















if u and v are orthogonal

ANGLE BETWEEN 2
VECTORS
v
u
v
u.
cos 

u
v

If the vectors lies on the same line or parallel to each other, then 0


u v u
v

Example 3
Find the angles between and
 
  6
,
0
,
4
3
,
0
,
2
4
,
3
,
2
3
,
2
,
1







v
u
ii
v
u
i
u v

Exercise :
a)
b)
c)
d)
Determine either this two vectors 2, 4,1 and 6,12, 3 are
parallel or not.
u v
    
 
Find the terminal point of 3 2 if the initial points is 1, 2
v i j
  
 
Given 3 2 , and 2 2 4 . Find
)
) 2
) 8 2
u i j k v i j w i j k
i w v
ii v w
iii v w u
       

 
 
Find the angle between the vector 2 2 and
) 3 6 2
) 2 7 6
u i j k
i v i j k
ii z i j k
  
   
  

DIRECTIONS OF ANGLES
& DIRECTIONS OF
COSINES





 ,
, - Are the angles that the vector OP
makes with positive axis
- Knows as the direction angles of
vector OP
DIRECTION OF COSINES







180
,
,
0
;
cos
cos
cos






OP
z
OP
y
OP
x

Example 1
Find the direction cosines and direction angles of:
 
     
3
,
2
,
1
1
,
1
,
0
3
,
1
,
2




Q
P
ii
u
i

CROSS PRODUCT
The result is a vector
If and then:
3
2
1 ,
, u
u
u
u  3
2
1 ,
, v
v
v
v 
     k
v
u
v
u
j
v
u
v
u
i
v
u
v
u
v
v
v
u
u
u
k
j
i
v
u
1
2
2
1
1
3
3
1
2
3
3
2
3
2
1
3
2
1









Example 3
Find the cross product between and
 
  6
,
0
,
4
3
,
0
,
2
4
,
3
,
2
3
,
2
,
1







v
u
ii
v
u
i
u v

CROSS PRODUCT
Properties of Cross Product
 
       
       
 
 
 
 
 
  j
k
i
i
j
k
k
i
j
ix
j
i
k
i
k
j
k
j
i
viii
k
k
j
j
i
i
vii
v
u
v
u
vi
v
u
v
a
u
v
i
v
k
u
v
u
k
v
u
k
iii
w
u
v
u
w
v
u
ii
u
v
v
u
i











































,
,
,
,
0
sin
0
0
0
0
 if u and v are parallel

APPLICATIONS
Equation of Planes
P0(x0,y0,z0) P(x,y,z)
n
Vector is on the plane M and vector which is perpendicular to M
known as normal vector, n
P
P0
From the properties of Dot Product
0
.
0 
n
P
P

APPLICATIONS
Equation of Planes
v
u
n 

Normal vector n :
u
v
v
u
n 


APPLICATIONS
Let and
 
0
0
0
0 ,
,
,
,
, z
y
x
P
c
b
a
n   
z
y
x
P ,
,
d
cz
by
ax
cz
by
ax
cz
by
ax
c
b
a
z
z
y
y
x
x
n
P
P













0
0
0
0
0
0
0
0
,
,
.
,
,
0
.
EQUATION OF PLANE

Example 6:
Find an equation of plane through P(-3,0,7) perpendicular to
Find an equation of plane through P(2,-5,-1) perpendicular to
1
,
2
,
5 

n
1,4,2
n  4 2 20 0
x y z
   

Example 7:
Find equation of plane through 3 points:
     
2,1,1 0,2,3 1,0, 1
 
2 1
y z
 

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