Overview

() Overview

(Chapter 1)

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Chapter (1)

1.6

SI UNIT

1.3

SYSTEM

1.2

VECTORS

Outline

WORKING WITH NUMBERS

What is a Vector ? (Difference between vector and scalar quantity)

Cartesian Coordinate System (2D &3D)

Vector addition using components

Multiplication of Vectors with a Scalar

Unit Vector (write a vector in a unit vector notation)

Vector Length and Direction

Vectors multiplication

(Scalar and vector product) Resolve vectors and find their components

After studying this chapter, you will be able to:

1. Convert decimal numbers to and from scientific notation using positive and negative exponents.

2. Recognize the SI units.

3. Identify and use Prefixes.

4. Convert between units.

5. Understand the Cartesian coordinate system.

6. Define vector quantity.

7. Differentiate between vector and scalar quantities.

8. Resolve any vector and find its components.

9. Calculate the magnitude and direction of vectors.

10. Identify the unit vectors ( magnitude and direction) on three axes.

11. Write a vector in a unit vector notation.

12. Add vectors by components.

13. Multiply vectors by a scalar (either +ve or - ve no.).

14. Calculate the scalar product of two vectors.

15. Calculate the cross product of two vectors.

Learning Outcomes

Working with numbers (Scientific Notation)

❑ Dealing with really big numbers or really small numbers can be difficult.

To deal with big and small numbers we use scientific notation:

Number = mantissa x 10^exponent

⮚ The mantissa is usually chosen so that it has one digit preceding the decimal point, but not always.

e.g. 3.00x10⁸.

❑ Multiplication and division are simplified using scientific notation:

e.g. (7x10²⁷)(7x10⁹) = 49x10³⁶ 1.2

3560000000.0 m 3.56x10

⁺⁹

m

0.00000492 s 4.92x10

^-6

s

❑ Big number:

❑ Small number:

Extra Exercise

35.6x10⁺⁸ m

0.492x10^-5 s 49.2x10^-7 s

Express 0.00592 in scientific notation.

a) 5.92 × 10

³

b) 5.92 × 10

⁻³

c) 5.92 × 10

⁻²

d) 5.92 × 10

⁻⁵

e) 5.92 × 10

⁵

Extra Exercise

SI Unit System

■ The international system of units is abbreviated SI:

– Used for scientific work around the world.

■ The seven base units are:

Based on the General Conference on Weight and Measurements in 1971

1.3

Derived Units

■ Units for all other physical quantities are derived from the seven base units.

– For example: velocity =length/time=m/s

■ Some derived units that are used often are given names.

– Often named for famous physicists

– For example: Force=mg= kg m/s^{2 =} newton (N) 1.3

Prefix

Prefix Symbol Factor

Tera- T 10¹²

Giga- G 10⁹

Mega- M 10⁶

Kilo- K 10³

Centi- c 10^-2

Milli- m 10^-3

Micro- μ 10^-6

Nano- n 10^-9

Pico- p 10^-12

❑ Prefix represents a certain power of 10, to be used as a

multiplication factor.

❑ Attaching a prefix to an (SI) unit has the

effect of multiplying by the associated factor 1.3

Prefix Symbol Factor

Tera- T 10¹²

Giga- G 10⁹

Mega- M 10⁶

Kilo- K 10³

Centi- c 10^-2

Milli- m 10^-3

Micro- μ 10^-6

Nano- n 10^-9

Pico- p 10^-12

Extra Exercise

Unit conversion

Prefix Symbol Factor

Tera- T 10¹²

Giga- G 10⁹

Mega- M 10⁶

Kilo- K 10³

Centi- c 10^-2

Milli- m 10^-3

Micro- μ 10^-6

Nano- n 10^-9

Pico- p 10^-12

1.3

A section of a river can be approximated as a rectangle that is 20 m wide and 30 m long. Express the area of this river in square

kilometers.

a) 600 km² b) 6 km²

c) 6 × 10⁻² km² d) 6 × 10⁻⁴ km² e) 6 × 10⁺⁴ km²

Extra Exercise

On the Y- axis

origin 0 Positive direction

Negative direction

●

Cartesian Coordinate System

1.6

P = (3, 4)

P = (3, 4, 3)

Cartesian Coordinate System

1.6

PHYSICAL QUANTITIES

scalar Quantites

Vector quantities

magnitude

Example

Speed

Temperature time

Example

magnitude

Displacement Velocity

What is a Vector ?

(Difference between vector and scalar quantities)

direction

1.6

What is a Vector?

1.6

▪ Resolving a vector is the process of finding its components.

▪ A component is the projection of the vector on an axis.

⮚ for example is the component of vector a on (or along) the x axis and , is the component along the y axis.

Resolve a vector and find its components

Measuring the angle from x direction Measuring the angle from y direction

1.6

A more general way of finding vector components the angle is measured with respect to x-axis

Resolve a vector and find its components

1.6

More general ways of finding vector component Angle is measured with respect to y-axis

Resolve a vector and find its components

1.6

Resolve a vector and find its components

1.6

How to express a vector

How to express a Vector?

Magnitude-angle notation (Vector length and direction)

Unit vector notation

vector components 1.6

𝑎 = 𝑎_𝑥 𝑥 + 𝑎_𝑦 𝑦 + 𝑎_𝑧 𝑧

 Note also that a vector 𝑎 can be written using the Cartesian components notation as follows:

As written in the textbook is

Another famous unit vector notation is

■ Knowing the components of a vector, we can calculate its length and direction.

■ Vectors in 2 dimensions (most important case)

Vector Length and Direction

The direction The length

(using Pythagorean theorem)

1.6

ϴ + ϴ -

1.6

Vector Length and Direction

Direction

Exercise 1.77 (Page 30)

Extra Exercise

• Writing a vector in magnitude - angle notation

• Finding the components.

Vector resolving

1.6

Vector Length and Direction

Unit Vectors (write a vector in the unit vector notation)

1.6

1.6

 Note for this representation : 𝑎 = 𝑎_𝑥 𝑥 + 𝑎_𝑦 𝑦 + 𝑎_𝑧 𝑧

𝑥 = (1,0,0) 𝑦 = (0,1,0) 𝑧 = (0,0,1) the unit vectors are:

Unit Vector notation

Extra Exercise

1) The unit vector notation is represented as:

71°

63°

■ Adding vectors using the component method:

Note: this can also be applied to vector subtraction

Vector Addition using Components

1.6

Vector x-component y-component

64.9 37.5

– 56.7 19.5

– 15.3 – 19.7

80.19 – 40.85

Exercise: 1.68 (Page 30)

+ +

+ +

+ +

73.09 – 3.55

71°

63°

Vector x-component y-component

64.9 37.5

– 56.7 19.5

80.19 – 40.85

Exercise: 1.68 (Page 30)

+ +

201.79 – 22.85

– –

71°

63°

b)

71°

63°

Multiplication of a vector with a positive scalar

results in another vector that points in the same direction but has a magnitude that is the product of the scalar and the magnitude of the original vector.

Multiplication of a vector with a negative scalar

results in another vector that points in the opposite direction but has a magnitude that is the absolute value of the product of the negative scalar and the magnitude of the original vector.

Multiplication of a Vector with a Scalar

1.6

Multiplying a vector by a vector

Scalar product

(or Dot product) Vector product

(or Cross product)

will produce a scalar

will produce a new vector

Multiplication of a Vector with a Vector

1.6

■ It is sometimes called the dot product.

■ The scalar vector product is defined as:

■ In terms of Cartesian coordinates:

■ If the two vectors form a 90°angle, the scalar product is zero.

■ We can use the scalar product to find the angle between two vectors in terms of their Cartesian coordinates:

Scalar Product (dot product)

1.6

Exercise: 1.15 (Page 28)

Answer:

The two vectors are given in Cartesian Coordinate, and we do not have the angle between the two vectors, hence we will use;

Example: 1.5 (Page 23)

Vector product(cross product)

1.6

𝐶 = 𝐴 × 𝐵 =

𝑥 𝑦 𝑧 𝐴_𝑥 𝐴_𝑦 𝐴_𝑧

𝐵_𝑥 𝐵_𝑦 𝐵_𝑧 = 𝐴_𝑦𝐵_𝑧 − 𝐴_𝑧𝐵_𝑦 𝑥 − 𝐴_𝑥𝐵_𝑧 − 𝐴_𝑧𝐵_𝑥 𝑦 + (𝐴_𝑥𝐵_𝑦− 𝐴_𝑦𝐵_𝑥) 𝑧

Alternatively, we can simply use the determinant method

Vector product(cross product)

1.6

Exercise

1.6

𝑐 = 𝑎 × 𝑏 = 𝑖 𝑗 𝑘

3 −4 0

−2 0 3

= −4 × 3 − 0 × 0 𝑖 − 3 × 3 − 0 × −2 𝑗 + 3 × 0 − −2 × −4 𝑘

= −12 𝑖 − 9 𝑗 − 8 𝑘

Equation summary

Vector Notation

Vector Length and direction

Vector resolving, angle is measured from the xdirection

in first quadrant Dot

product

Angle between two vectors Vector resolving,

angle is measured from the ydirection in first quadrant

Vector product

𝐴 × 𝐵 =

𝑥 𝑦 𝑧 𝐴_𝑥 𝐴_𝑦 𝐴_𝑧 𝐵_𝑥 𝐵_𝑦 𝐵_𝑧

THE END CHAPTER OF

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