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1. SCALAR AND VECTOR
SCALAR AND VECTOR

2. Scalars are quantities which have
magnitude without direction
Examples of scalars
• temperature
• mass
• kinetic energy
• time
• amount
• density
• charge
Scalars

3. Vector
A vector is a quantity that has both
magnitude (size) and direction
it is represented by an arrow whereby
– the length of the arrow is the magnitude, and
– the arrow itself indicates the direction
The symbol for a vector is a letter
with an arrow over it
A
Example

4. Two ways to specify a vector
It is either given by
• a magnitude A, and
• a direction 
Or it is given in the
x and y components as
• Ax
• Ay
y
x

A
A
y
x
Ax A
Ay

5. y
x
Ax A
Ay
A

x
A = Acos
Ay = Asin
│A │ =√ (A 2
+A 2
)
x y
Themagnitude(length) of Aisfound byusingthe
PythagoreanTheorem
Thelengthof avector
clearlydoesnot
dependonitsdirection.

6. y
x
Ax A
Ay
A

Thedirectionof Acanbestatedas
tan =Ay/ Ax
 =tan-1(Ay/ Ax)

7. Some Properties of Vectors
Equalityof TwoVectors
Twovectors AandBmay bedefined tobe
equal if theyhavethesamemagnitudeand
point inthesamedirections. i.e. A=B
A B
A
A
B
B

8. Negativeof aVector
Thenegativeof vector Aisdefinedasgivingthe
vector sumof zerovaluewhenaddedtoA.
That is, A+(- A) =0. Thevector Aand –A
have thesamemagnitudebut areinopposite
directions.
A
-A

9. Scalar Multiplication
The multiplication of a vector A
by a scalar 
- will result in a vector B
B =  A
- whereby the magnitude is changed
but not the direction
• Do flip the direction if  is negative

10. B =  A
If  =0, therefore B =  A=0,
which isalsoknownasazerovector
(A) = A = (A)
(+)A = A + A
Example

11. The addition of two vectors A and B
- will result in a third vector C called the resultant
A
B
C
C = A + B
Geometrically (triangle method of addition)
• put the tail-end of B at the top-end of A
• C connects the tail-end of A to the
top-end of B
Wecanarrangethevectors aswelike,aslong
aswemaintaintheirlengthanddirection
Vector Addition
Example

12. More than two vectors?
x1
x5
x4
x3
x2
xi
xi =x1 +x2 +x3+x4+x5
Example

13. Vector Subtraction
Equivalent to adding the negative vector
A
-B
B
C= A B
C= A+(-B)
Example

14. Rules of Vector Addition
commutative
A + B = B + A
A
B
B
A

15. associative
(A + B) + C = A + (B + C)
B
C
A
B
C
A A+B
(A+B)+C
A+(B+C)
B+C

16. distributive
m(A + B) = mA + mB
A
B
A+B mA
mB
m(A+B)

17. Parallelogram method of addition
(tailtotail)
A
A+B
B
The magnitude of the resultant depends on the
relative directions of the vectors

18. 


Unit Vectors


a vector whose magnitude is 1 and
dimensionless
the magnitude of each unit vector equals
a unity; that is, │
i │=j│ │k
= │ │= 1
and defined as
i a unit vector pointing in the x direction
j a unit vector pointing in the y direction
k a unit vector pointing in the z direction


19. Useful examples for the Cartesian
unit vectors [ i, j, k ]
- they point in the direction of the
x, y and z axes respectively
x
z
i
y
j
k

20. Component of a Vector in 2-D
x-axis
vector A can be resolved into two components
Ax and Ay
y-axis
Ay
Ax
A
θ
A=Ax +Ay

21. The component of A are
│Ax│=Ax=Acosθ
│Ay│=Ay=Asin θ
The magnitude of A
A = √Ax + Ay
2 2
The direction of A
tan =Ay/ Ax
 =tan-1(Ay/ Ax)
x-axis
y-axis
Ay
Ax
A
θ
Example

22. The unit vector notation for the vector A
x-axis
Ax
is written
A=Axi +Ayj
y-axis
Ay
A
θ
i
j
Example

23. Component of a Vector in 3-D
A
Ay
Az
vector A can be resolved into three components
Ax , Ay and Az
z-axis
y-axis
i
Ax
x-axis
j
k
A=Axi +Ayj +Azk

24. if
A=Axi +Ayj +Azk
B=Bxi +Byj +Bzk
A+B=C sumof thevectorsAandBcan
thenbeobtainedasvector C
C=(Axi +Ayj +Azk)+(Bxi +Byj+Bzk)
C=(Ax +Bx)i+(Ay +By)j +(Az +Bz)k
C=Cxi +Cyj +Czk Example

25. Dot product (scalar) of two vectors
Thedefinition:
θ
B
A
A· B=│A││B│cosθ

26. Dot product (scalar product) properties:
if θ = 900 (normal vectors) then the dot
product is zero
|A· B|=ABcos90=0 and i · j = j · k = i · k = 0
if θ = 00 (parallel vectors) it gets its maximum
value of 1
|A· B|=ABcos0=1 and i · j = j · k = i · k = 1

27. A + B = B + A
the dot product is commutative
Use the distributive law to evaluate the dot product
if the componentsare known
A· B=(Axi +Ayj +Azk) ·(Bxi +Byj +Bzk)
A. B =(AxBx) i.i +(AyBy) j.j +(AzBz) k.k
A. B=AxBx +AyBy +AzBz
Example

28. Cross product (vector) of two vectors
Themagnitudeof thecrossproduct givenby
thevectorproductcreatesanewvector
thisvector isnormaltothe planedefinedbythe
originalvectorsanditsdirectionisfoundbyusingthe
right handrule
│C│=│AxB│=│A││B│sinθ
θ
A
B
C

29. Crossproduct (vector product) properties:
if θ = 00 (parallel vectors) then the cross
product is zero
|AxB|=ABsin0=0 and i x i = j x j = k x k = 0
if θ = 900 (normal vectors) it gets its maximum
value
|AxB|=ABsin90=1 and i x i = j x j = k x k = 1

30. the relationship between vectors i , j and k can
be described as
i x j = - j x i = k
j x k = - k x j = i
k x i = - i x k = j
Example

31. THEEND