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

������������������������������������������

��������������

������������������������������������������

��������������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

��������������

������������������������������������������

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