Units, Physical Quantities, and Vectors

Introduction

This chapter introduces the foundational concepts of physical quantities, units, and vectors, which are essential for understanding and solving problems in physics with calculus. Mastery of these topics is crucial for all subsequent chapters in university-level physics.

Scalars and Vectors

Definitions and Distinctions

• Scalar Quantity: A physical quantity described by a single number (magnitude) and appropriate units. Examples: mass, temperature, time.

• Vector Quantity: A physical quantity that has both magnitude and direction. Examples: displacement, velocity, force.

• Vectors are typically represented in boldface with an arrow, e.g., \( \vec{A} \). The magnitude is written as A or |\( \vec{A} \)|.

Drawing and Types of Vectors

Graphical Representation

• Vectors are drawn as arrows: the length represents magnitude, and the arrowhead indicates direction.

Types of Vectors

• Unit Vector: A vector with magnitude 1, indicating direction only.

• Equal Vectors: Vectors with the same magnitude and direction.

• Negative Vector: A vector with the same magnitude but opposite direction.

• Parallel Vectors: Vectors that have the same or exactly opposite direction.

• Resultant Vector: The vector sum of two or more vectors.

• Equilibrant Vector: The vector that balances other vectors to produce equilibrium.

Addition and Subtraction of Vectors

Graphical Methods

• Triangle Law: Place vectors head-to-tail; the resultant is from the tail of the first to the head of the last.

• Parallelogram Method: Place vectors tail-to-tail and complete the parallelogram; the diagonal is the resultant.

Properties of Vector Addition

• Vector addition is commutative: \( \vec{A} + \vec{B} = \vec{B} + \vec{A} \).

• Vector addition is associative: \( (\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C}) \).

Adding More Than Two Vectors

• Use the head-to-tail method for multiple vectors; the resultant is independent of the order of addition.

Subtracting Vectors

• Subtracting \( \vec{B} \) from \( \vec{A} \) is equivalent to adding \( -\vec{B} \) to \( \vec{A} \).

Multiplying a Vector by a Scalar

• Multiplying by a positive scalar changes the magnitude but not the direction.

• Multiplying by a negative scalar reverses the direction.

Addition of Two Vectors at Right Angles

• When vectors are perpendicular, use the Pythagorean theorem to find the resultant magnitude:

• The direction is found using trigonometry:

Components of a Vector

Resolving Vectors into Components

• Any vector can be expressed as the sum of its x- and y-components:

• Where and

Positive and Negative Components

• Components can be positive or negative depending on the vector's direction relative to the axes.

Calculations Using Components

• To find the magnitude and direction from components:

Finding Components with Angles

• Be careful with the angle's reference direction; use the correct sign for each component.

Unit Vectors

Definition and Use

• Unit vectors have magnitude 1 and indicate direction along coordinate axes: (x), (y), (z).

• Any vector can be written as:

Scalar (Dot) Product of Vectors

Definition and Properties

• The scalar product (dot product) of two vectors is:

• Where is the angle between the vectors.

• The result is a scalar (not a vector).

Sign of the Scalar Product

• Positive if

• Negative if

• Zero if

Calculating the Scalar Product Using Components

• For vectors and :

Finding an Angle Using the Scalar Product

• To find the angle between two vectors:

Vector (Cross) Product of Vectors

Definition and Properties

• The vector product (cross product) of two vectors is:

• Where is the angle between the vectors and is a unit vector perpendicular to the plane of and (direction given by the right-hand rule).

• The result is a vector.

Anticommutativity of the Vector Product

Calculating the Vector Product

• For vectors and :

Summary Table: Vector Operations

Additional info: These notes provide a comprehensive overview of vectors, their properties, and operations, which are foundational for all topics in calculus-based physics. Mastery of these concepts is essential for understanding motion, forces, energy, and more in later chapters.