chapter 3 : vectors - introduction - addition of vectors - subtraction of vectors - scalar...
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Chapter 3 : Vectors
- Introduction- Addition of Vectors- Subtraction of Vectors- Scalar Multiplication of Vectors- Components of Vectors- Magnitude of Vectors- Product of 2 Vectors- Application of Scalar/Dot Product & Cross Product
Introduction
• Has magnitude (represent by length of arrow) .
• direction (direction of the arrow either to the right, left, etc).
• Eg: move the brick 5m to the right
Vectors
Scalars• Has magnitude
only.• Eg: move the brick
5m.
Introduction
• Use an arrow connecting an initial point A to terminal point B.
• Denote
• Written as • Magnitude of
Vectors Representation
AB��������������
AB AB����������������������������
Introduction
• Vector in opposite direction, , but has same magnitude as .
Vectors Negativea a
Introduction
• If we have 2 vectors, with same magnitude & direction .
Equal Vectors
Addition of Vectors
• Any 2 vectors can be added by joining the initial point of to the terminal point of .
• Eg:
1. The Triangle Lawb
a
Addition of Vectors
• If 2 vector quantities are represented by 2 adjacent sides of a parallelogram, then the diagonal of parallelogram will be equal to the summation of these 2 vectors.
• Eg:
• The parallelogram law is affected by the triangle law.
2. The Parallelogram Law
Addition of VectorsThe sum of a number of vectors
Subtraction of Vectors
• Is a special case of addition.
• Eg:
Scalar Multiplication
• k ; vector multiply with scalar, k.
• .
a a
Parallel Vectors
Parallel Vectors
Scalar Multiplication
Components of Vectors – Unit Vectors
Vectors in 2 Dimensional (R2)
Vectors in 3 Dimensional (R3)
Exercise :
Draw the vector
i. 2 6
ii. 4 5 2
i j
i j k
Components of Vectors
Magnitude of Vectors
Exercise:
Example:
1. For Any Vector
Magnitude of Vectors
Example:
2. From one point to another point of vector
- point / coordinate
vector
Magnitude of VectorsSolution:
2 2 2
i) P to Q = = 9 1, 2 5, 4 7 = 8, 3, 3
8 ( 3) ( 3) 82
ii) Q to R = = 3 9, 2 2, 6 4 = 6,0,2
PQ OQ OP
PQ
QR OR OQ
QR
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��������������
������������������������������������������
��������������2 2( 6) 0 2 40
Do Exercise 3.3 in Textbook page 70.
Unit Vectors
Example:
Do Exercise 3.4 in Textbook page 70.
Direction Angles & Cosines
, , : direction angles of vector OP ��������������
cos ,cos ,cos : direction cosines of the vector
cos ,cos ,cos
OP
x y z
OP OP OP
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Direction Angles & CosinesExample:
Solution (i):Direction cosines
Direction angles
90.77
Direction Angles & CosinesSolution (ii)
2 2 2
= 3 5, 4 7, 1 2 = 8, 3,3
( 8) ( 3) 3 82
8 3 3cos ,cos ,cos ,
82 82 82
PQ OQ OP
PQ
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��������������
Direction cosines
Direction angles 1
1
1
8cos 152.06
823
cos 109.35823
cos 70.6582
Do Exercise 3.5 in Textbook page 72.
Do Tutorial 3 in Textbook page 85 :
•No. 2 (i)•No. 3 (i)
•No. 4•No. 5 (iii)•No. 6 (i)
Operations of Vectors by Components
Example:
Solution:
Do Exercise 3.6 in Textbook page 72.
Product of 2 Vectors
Example:
Solution:
Dot Product / Scalar Product
Do Exercise 3.7 in Textbook page 73.
Find Angle Between 2 Vectors
Example:
Solution:
Do Exercise 3.8 in Textbook page 74.
Product of 2 Product
Example:
Cross Product / Vector Product
Product of 2 ProductCross Product / Vector Product
Solution:
i) 4 7 1 (35 1) (20 2) (4 14)
2 1 5 =36 22 10
ii) 2 1 5 ( 1 35) ( 2 20) (14 4)
4 7 1 =-36 22 10
i j k
u v i j k
i j k
i j k
v u i j k
i j k
Do Exercise 3.9 in Textbook page 74.
Find Angle Between 2 Vectors
Applications of Vectors
• Projections• The Area of Triangle & Parallelogram• The Volume of Parallelepiped & Tetrahedron• Equations of Planes
• Parametric Equations of Line in R3
• Distance from a Point to the Plane
i. Projections
Scalar projection of b onto a:
Vector projection of b onto a:
..a
a b acomp b b scalar
a a
.a a
a b a aproj b comp b vector
a a a
Example :
i.Given . Find the scalar projection and vector projection of b onto a
ii.Find given that
Solutions:
2 3 and 2 3a i j k b i j k
and a acomp b proj b 4 3 and 2a i j k b i j k
ii. The Area of Triangle and Parallelogram
Area of triangle POQ = 1/ 2 sin 1/ 2
Area of parallelogram OQRP sin
Note that parallelogram can be divided into 2 triangles.
a b a b
a b a b
Example :
Solutions:
Solutions:
iii. The Volume of Parallelepiped and Tetrahedron
A parallelepiped is a three-dimensional formed by six parallelogram.
•Define three vectors•To represent the three edges that meet at one vertex. •The volume of the parallelepiped is equal to the magnitude of their scalar triple product
1 2 3 1 2 3 1 2 3, , , , , , , ,a a a a b b b b c c c c
V a b c
•Volume of Parallelepiped
•Volume of Tetrahedron
=
V a b c
b c a
c a b
1 2 3
1 2 3
1 2 3
1
6
a a a
V a b c b b b
c c c
Example :
Solution:
iv. Equations of Planes
Example:
Solutions:
Example :
Solutions:
v. Parametric Equations of a Line in 3R
Parametric equations of a line :
Cartesian equations :
Example :
Solutions:
vi. Distance from a Point to the Plane
Example:
Solutions:
ii.1
2
2 2 2
n 10,2, 2
n 5,1, 2
Let 1st equation to find the point
Let x=z=0
10(0) 2 2(0) 5
5
25
(0, ,0)2
50(5) (1) 0( 2) 1
2 0.28875 1 ( 2)
Vector
Vector
y
y
P
D