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Vector
1. GROUP’S MEMBERS
Name Matric No.
Ridwan bin shamsudin D20101037472
Mohd. Hafiz bin Salleh D20101037433
Muhammad Shamim Bin D20101037460
Zulkefli
Jasman bin Ronie D20101037474
Hairieyl Azieyman Bin Azmi D20101037426
Mustaqim Bin Musa D20101037402
4. WHAT IS VECTOR?
VECTOR REPRESENTATIVE
MAGNITUDE OF VECTOR
NEGATIVE VECTOR
ZERO VECTOR
EQUALITY OF VECTOR
PARALLEL VECTOR
VECTOR MULTIPLICATION BY
SCALAR
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7. INTRODUCTION . .
Vectoris a variable quantity that can
be resolved into components.
Vector also is a straight line segment
whose length is magnitude and whose
orientation in space is direction.
Hurmm . . .
8. SCALAR VECTOR VECTOR PRODUCT
A scalar quantity
has magnitude only A vector quantity has
with an appropriate both magnitude and
unit of direction.
measurement. Examples of vector
Example of scalar
quantities are quantities are
length, speed, time, displacement,
temperatue, mass velocity, acceleration
and power. and force.
9. The most commonly used example of vectors in
everyday life is velocity.
Vectors also mainly used in physics and engineering
to represent directed quantities.
Vectors play an important role in physics about of a
moving object and forces acting on it are all
described by vectors.
11. Since several important physical
quantities are vectors, it is useful to
agree on a way for representing them
and adding them together.
In the example involving displacement,
we used a scale diagram in which
displacements were represented by
arrows which were proportionately
scaled and orientated correctly with
respect to our axes (i.e., the points of the
compass).
12. Thisrepresentation can be used for all vector
quantities provided the following rules are
followed:
1.The reference direction is indicated.
2.The scale is indicated.
3.The vectors are represented as arrows
with a length proportional to their
magnitude and are correctly orientated with
respect to the reference direction.
4.The direction of the vector is indicated by
an arrowhead.
5.The arrows should be labelled to show
which vectors they represent.
13. For example, the diagram below
shows two vectors A and B, where A
has a magnitude of 3 units in a
direction parallel to the reference
direction and B has a magnitude of 2
units and a direction 60° clockwise to
the reference direction:
I see ~
14.
15. The length of a vector is called the
magnitude or modulus of the vector.
A vector whose modulus is unity is
called a unit vector which has
magnitude. The unit vector in the
direction is called
The unit vectors parallel to
16. The magnitude of vector a is written as |a|.
The magnitude of vector AB is written as
|AB|. 𝑥 𝑥2 + 𝑦2
𝑦
If a = then the magnitude |a|=
*using pythagorean theorem.
19. A vector having the same magnitude
but opposite direction to a vector A, is
-A.
If v is a vector, then -v is a vector
pointing in the opposite direction.
If v is represented by (a, b, c)T then -v
is represented by (-a, -b, -c)T.
20.
21. Example :
Write down the negetive of
solution :
3
= −
−2
−3
= 2
22.
23.
24. • Is a vector with zero magnitude and
no direction
• |0|= 0
25. EXAMPLE :
Determine whether w-y-x+z is a zero
vector.
Solution
From the diagram,w-y-x+z = O
Since it does not has magnitude,thus it is
a zero
vector
26.
27. 2 vectors u and v are equal if their
corresponding components are equal
For example,
if u=ai +bj and v=ci + dj
then u = v a=c and b=d
Or in another word we can say it is
equal if the vectors have same
magnitude and same direction
30. Vectors are parallel if they have the
same direction
Both components of one vector must
be in the same ratio to the
corresponding components of the
parallel vector.
(i) v1 kv2 , k any scalar
(ii) v1 .v2 v1 v2 or
v1 .v2 v1 v2
v x v 0
(iii) 1 2
31. EXERCISE
Exercise
Given 2i-3j and 8i+yj are parallel vector.
Find the value of y.
Solution
Since they are parallel vectors
Let 8i+yj=k(2i-3j),k is any scalar
8i+yj=2ki-3kj
8=2k y=-3k
k=4 =-3(4)
=-12
34. The scalar product(dot product) of two
vectors and is denoted by
and defined as
a b a b cos
Where is the angle between and
which converge to a point or diverge from
a
point.
44. 1) a . a = a 2
2) a . b = b . a
3) a . (b + c) = a . b + a . c
4)
(a b )c) (a b c) a b c
5) m (a . b) = (ma) . b = (a . b)m
6) a . b = a b if and only if a parallel to b
a . b = – a b if and only if a and b in
opposite direction
7) a . b = 0 if and only if a is perpendicular to b
8)
.
45. Example
:
Evaluate
a) (2 i j ) (3 i 4 k )
~ ~ ~ ~
b) (3 i 2 k ) (i 2 j 7 k )
~ ~ ~ ~ ~
50. More about Vector
Multiplication
There are three different types of
multiplication: dot product, cross product,
and multiplication of vector by a scalar.
The dot product of two vectors u and v is
given as u · v = uv cos θ where θ is the
angle between the vectors u and v.
The cross product of two vectors u and v
is given as u × v = uv sin θ where θ is
the angle between the vectors u and v.
When a vector is multiplied by a scalar,
only the magnitude of the vector is
changed, but the direction remains the
same.
51. Examples of Vector
Multiplication
If the vector is multiplied by a scalar then
=.
If u = 2i + 6j and v = 3i - 4j are two
vectors and angle between them is 60°,
then to find the dot product of the
vectors, we first find their magnitude.
Magnitude of vector
Magnitude of vector
The dot product of the vectors u, v is u ·
v = uv cos θ
= (2 ) (5) cos 60°
= (2 ) (5) ×
=5
52. If u = 5i + 12j and v = 3i + 6j are two
vectors and angle between them is 60°,
then to find the cross product of the
vectors, we first find their magnitude.
Magnitude of vector
Magnitude of vector
The cross product of the vectors u, v is u
× v = uv sin θ
= (3 ) (13) sin 60°
= 39 (2)
= 78
53. Solved Example on Vector Multiplication
Which of the following is the dot product of the
vectors u = 6i + 8j and v = 7i - 9j?
Choices:
A. 114
B. - 30
C. - 2
D. 110
Correct Answer: B
Solution:
Step 1: u = 6i + 8j, v = 7i - 9j are the two vectors.
Step 2: Dot product of the two vectors u, v = u · v
= u1v1 + u2v2
Step 3: = (6i + 8j) · (7i - 9j)
Step 4: = (6) (7) + (8) (- 9) [Use the definition of
the dot product of two vectors.]
Step 5: = - 30 [Simplify.]
54.
55. Definition of Addition of
Vectors
Adding two or more vectors to form a
single resultant vector is known as
Addition of Vectors.
56. More about Addition of
Vectors
If two vectors have the same direction,
then the sum of these two vectors is
equal to the sum of their magnitudes,
in the same direction.
If the two vectors are in opposite
directions, then the resultant of the
vectors is the difference of the
magnitude of the two vectors and is in
the direction of the greater vector.
57. Examples of Addition of
Vectors
To find the sum of the vectors of and , they
are placed tail to tail to form two adjacent
sides of a parallelogram and the diagonal
gives the sum of the vectors and . This is
also called as ‘parallelogram rule of vector
addition’.
58. If the vector is represented in
Cartesian coordinate, then the sum of
the vectors is found by adding the
vector components.
The sum of the vectors u = <- 3, 4>
and v = <4, 6> is u + v =
<- 3 + 4, 4 + 6>
= <1, 10>
59.
60. Definition Of Subtraction Of
Vectors
subtracting two or more vectors to
form a single resultant vector is known
as subtraction of vectors.
61. example
f the vector is represented in
Cartesian coordinate, then the
subtraction of the vectors is found by
subtracting the vector components.
The sum of the vectors u = <- 3, 4>
and v = <4, 6> is u - v =
<- 3 - 4, 4 - 6>
= <-7, -2>
62.
63.
64. The angle between 2 lines
The two lines have the equations r = a
+ tb and r = c + sd.
The angle between the lines is found
by working out the dot product of b
and d.
We have b.d = |b||d| cos A.
65. Example
Find the acute angle between the lines
L : r i 2 j t (2i j 2k )
1
L : r 2i j k s(3i 6 j 2k )
2
Direction Vector of L1, b1 = 2i –j + 2k
Direction Vector of L2, b2 = 3i -6j + 2k
If θ is the angle between the lines,
(2i j 2k ).( 3i 6 j 2k )
Cos θ =
2i j 2k 3i 6 j 2k
66. EXAMPLE
664
Cos θ =
9 49
16
Cos θ =
21
θ = 40 22’
