Study Notes on Scalar and Vector Quantities

Chapter Three: Scalar and Vector Quantities

Introduction

• Author: Zhwan M. Rashid

• Course: General Physics I (CMPE 171)

• Week: 2

• Semester: Fall 2025

Key Topics

• Coordinate systems

• Cartesian coordinate systems (2D)

• Plane polar coordinate system (2D)

• Conversion between the two systems

• Math review

• Trigonometry

• Pythagorean theorem

• Cosine law

• Sines law

• Scalar and vector quantities

  • Scalar quantities

  • Vector quantities

  • Components of a vector

  • Unit vectors

  • Vector applications

  • Adding vectors

  • Subtracting vectors

  • Vector multiplication

Learning Objectives

• Apply Cartesian and Plane Polar coordinate systems

• Employ Pythagorean's Theorem, Law of Sines, and Law of Cosines for mathematical problems

• Understand scalar and vector quantities, including decomposing vectors into components

• Explore mathematical applications of vector quantities, including multiplication of vectors

Coordinate Systems

Definition

• A coordinate system is a framework for identifying each point uniquely in a given space and is an artificial mathematical tool used to describe the position of a real object.

• A coordinate system consists of:

  • Origin: A fixed reference point (e.g., (0,0) in 2D)

  • Axes: Specific axes with scales and labels (e.g., x-axis, y-axis)

  • Labeling Instructions: How to label points relative to the origin and axes.

Types of Coordinate Systems

1. 1D Coordinate Systems

   • Simplest type, used for describing locations along a straight line (example: a train on an East-West track).

   • Position is described by a single real number (e.g., x-direction).

2. 2D Coordinate Systems

   • Require two numbers to define positions along two axes (x and y).

   • Examples include:

     • Cartesian Coordinate System: Points labeled as (x, y)

     • Polar Coordinate System: Points labeled as (r, θ)

3. 3D Coordinate Systems

   • Requires three numbers to describe position in three dimensions, adding a z-axis perpendicular to both x and y.

   • Examples include:

     • Cylindrical Coordinate System

     • Spherical Coordinate System

Plane Polar Coordinate System (2D)

• Origin and reference line define the system.

• Points are represented as (r, θ), with r being the distance from the origin at angle θ.

• Values related to x and y axes can be determined as follows:

  • For given x and y, calculate r:
    $r = ext{sqrt}(x^2 + y^2)$

  • The angle θ is calculated using trigonometric functions based on the coordinates:

  • heta = an^{-1}igg( rac{y}{x}igg)

Mathematical Review

Trigonometry

• Key identities:

  • Hypotenuse Relation:

  • $ext{cos} heta = rac{ ext{adjacent side}}{ ext{hypotenuse}}$

  • $ext{tan} heta = rac{ ext{opposite side}}{ ext{adjacent side}}$

  • Pythagorean Theorem:

  • $c^2 = a^2 + b^2$

Law of Cosines

• Useful for determining the third side of a triangle when two sides and the included angle are known:

  • $c^2 = a^2 + b^2 - 2ab ext{cos}(C)$

• Example for calculation:

  • Given angle C = 37°, sides a = 8, b = 11:

    • Calculate c using the formula:

    • $c^2 = 8^2 + 11^2 - 2 imes 8 imes 11 imes ext{cos}(37°)$

    • Result: $c ext{ (to 2 decimal places)} = 6.67$

Scalars and Vectors

Scalar Quantities

• Scalars have only magnitude, no direction.

• Examples include:

  • Speed: 20 m/s

  • Distance: 10 m

  • Age: 15 years

  • Heat: 1000 calories

Vector Quantities

• Vectors possess both magnitude and direction.

• Examples include:

  • Velocity: 20 m/s, North

  • Acceleration: 10 m/s², East

  • Force: 5 N, West

• Vectors are represented graphically by arrows, indicating direction and magnitude.

Properties of Vectors

• Equality of Two Vectors: Two vectors are equal if they have the same magnitude and direction.

• Negative Vectors: Vectors are negative if they have the same magnitude but are 180° apart.

Components of a Vector

• x-Component: Projection along the x-axis.

• y-Component: Projection along the y-axis.

• Components can be calculated as follows:

  • $A_x = A ext{cos}( heta)$

  • $A_y = A ext{sin}( heta)$

• For 2D vectors, the magnitude can be combined:

  • $A = ext{sqrt}(A_x^2 + A_y^2)$

  • Angle: heta = an^{-1}igg( rac{A_y}{A_x}igg)

Vector Applications

Vector Addition and Subtraction

• When adding vectors, direction must be considered, and units must be consistent.

• Example of Addition:

  • A man walks 54.5 m East, then 30 m East:

  • Displacement = 54.5 m + 30 m = 84.5 m, East

• Example of Subtraction:

  • Walking 54.5 m East then 30 m West:

  • Displacement = 54.5 m - 30 m = 24.5 m, East

Vector Representation and Manipulation

Perpendicular Vectors

• If vectors are perpendicular, the Pythagorean theorem can be applied:

  • Example calculation involving an angle and side lengths,

  • Use trigonometric functions to resolve into components.

Example Calculations

• Calculating resultant vectors from movements in two or three dimensions,

• Incorporation and usage of trigonometric functions to evaluate angles and components.

Scalar and Vector Products

Dot Product

• ext{A} ullet ext{B} = | ext{A}|| ext{B}| ext{cos}( heta) (results in a scalar)

Cross Product

• $ext{A} imes ext{B} = | ext{A}|| ext{B}| ext{sin}( heta)$ (results in a vector)

Conclusion

• Understanding of vector quantities is essential for problem-solving in physics,

• Scalars and vectors are used in various mathematical approaches to find resultant quantities and solve physical problems.

References

• Young, H. D., Freedman, R. A., & Ford, A. L. (2014). University physics with modern physics. New York: Pearson.

• Sommerfeld, A. (2016). Mechanics: Lectures on theoretical physics, Vol. 1. Elsevier.