Scalars
Scalars are quantities which have
magnitude without direction
Examples of scalars
• temperature • time
• mass • amount
• kinetic energy • density
• charge
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
A
with an arrow over it
Two ways to specify a vector
y
It is either given by A A
• a magnitude A, and θ
•a direction θ x
Orit is given in the
x and y components as y Ax A
• Ax Ay
• Ay x
y Ax A Ax = Acosθ
A Ay
θ Ay = A sin θ
x
The magnitude (length) of A is found by using the
Pythagorean Theorem
│A │ =√ ( Ax+Ay) 2 2
The length of a
vector
clearly does not
depend on its directi
y Ax A
A Ay
θ
x
The direction of A can be stated as
tan θ = Ay / Ax
θ =tan-1(Ay / Ax)
Some Properties of Vectors
Equality of Two Vectors
Two vectors A and B may be defined to be
equalif they have the same magnitudeand
point in the same directions. i.e. A = B
A
A B A B
B
Negative of a Vector
The negative of vector A is defined as giving
thevector sum of zero value when added to A .
Thatis, A + (- A) = 0. The vector A and -A
havethe same magnitude but are in opposite
directions.
A
-A
Scalar Multiplication
The multiplication of a vector A
by a scalar α
- will result in a vector B
B = αA
- whereby the magnitude is changed
but not the direction
• Do flip the direction if α is negative
B = αA
If α = 0, therefore B = α A = 0, which
isalso known as a zero vector
β(αA) = αβA = α(βA)
(β+α)A = αA + βA
Example
Vector Addition
The addition of two vectors A and B
- will result in a third vector C called the resultant
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 C
top-end of B B
We can arrange the vectors as we like, as A
long as we maintain their length and direction Example
More than two vectors?
x4
Σxi x5
x3
x1 x2 Σxi = x1 + x2+x3+x4+x5
Example
Vector Subtraction
Equivalent to adding the negative vector
C= A B
C = A + (-B) B
A
-B
Example
Rules of Vector Addition
commutative
A+B=B+A
B A
A B
associative
(A + B) + C = A + (B + C)
B
B
B+C C
A A+B C A
A + (B + C)
(A + B) + C
distributive
m(A + B) = mA + mB
mB
B
m(A+B)
A+B mA
A
Parallelogram method of addition
(tailtotail)
A A+B
The magnitude of the resultant depends on
the relative directions of the vectors
Unit Vectors
a vector whose magnitude is 1 and
dimensionless
the magnitude of each unit vector equals
aunity; 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
Useful examples for the Cartesian
unit vectors [ i, j, k ]
- they point in the direction of the
x,y and z axes respectively
y
i x
k
z
Component of a Vector in 2-D
vector A can be resolved into two components
Ax and A y
y- axis
A
Ay
θ
x-axis
Ax
A = Ax + Ay
The component of A are
y- axis
│Ax│ = Ax = A cos θ
│Ay│ = Ay = A sin θ A
Ay
θ
Themagnitude of A
A = √Ax2 + Ay 2 Ax x-axis
The direction of A
tan θ = Ay / Ax
θ =tan-1(Ay / Ax)
Example
The unit vector notation for the vector A
is written
A = Axi + Ayj
y- axis
Ay
A
j
θ
i Ax x- axis
Example
Component of a Vector in 3-D
vector A can be resolved into three components
A , A and A
x y z
z- axis
Az
A= Axi + Ayj + Azk
k A
j Ay y-axis
i
Ax
x- axis
if
A = Axi + Ayj + Azk
B = Bxi + Byj + Bzk
A + B = C sum of the vectors A and B can
then be obtained as vector C
C = (Axi + Ayj + Azk) + (Bxi + Byj + Bzk)
C = (Ax + Bx)i+ (Ay + By)j + (Az + Bz)k
C = Cxi + Cyj + Czk Example
Dot product (scalar) of two vectors
Thedefinition:
A
A· B= │A││B │cos θ
θ
B
Dot product (scalar product) properties:
if θ = 900 (normal vectors) then the dot
product is zero
|A· B| = AB cos 90 = 0 and i · j = j · k = i · k = 0
if θ = 00 (parallel vectors) it gets its maximum
value of 1
|A · B| = AB cos 0 = and
1 i · j = j · k = i ·k = 1
the dot product is commutative
A+B=B+A
Use the distributive law to evaluate the dot product
ifthe components are 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
Cross product (vector) of two vectors
The magnitude of the cross product given by
│C │=│A x B│ = │A││B │sin θ C B
θ
the vector product creates a new vector
A
this vector is normal to the plane defined
the by
original vectors and its direction is found by
right hand rule
Cross product (vector product) properties:
if θ = 00 (parallel vectors) then the cross
product is zero
|Ax B| = AB sin 0 = 0and i x i = j x j = k x k = 0
if θ = 900 (normal vectors) it gets its maximum
value
|A x B| = AB sin 90 =and
1 ixi=jxj=kxk=1
the relationship between vectors i , j and k can
be described as
ixj =-jxi =k
jxk=-kxj=i
kxi=-ixk=j
Example
Measurement and Error
THE END
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