Chapter 1: Vectors – Models, Measurements & Vectors in Physics

Vectors and Scalars

Definitions and Distinctions

In physics, quantities are classified as either scalars or vectors based on whether they possess direction in addition to magnitude.

• Scalar Quantity: Completely specified by a single value with an appropriate unit and no direction.

  • Examples: Mass, Temperature, Time

  • Scalars can be positive or negative (e.g., temperature), but some are always positive (e.g., mass).

  • Scalars are manipulated using the rules of ordinary algebra.

• Vector Quantity: Completely specified by a number with an appropriate unit (the magnitude) plus a direction.

  • Examples: Force, Velocity, Displacement

  • Vectors are represented graphically by arrows; the length indicates magnitude, and the arrow points in the direction.

Vector Notation and Representation

Graphical and Symbolic Representation

• Vectors are denoted by boldface letters or with an arrow above (e.g., \( \vec{A} \)).

• The tail of the arrow marks the starting point, and the head marks the end point, indicating direction.

• The magnitude is the length of the arrow, and the direction is the angle it makes with a reference axis.

Vectors in Two Dimensions

Magnitude and Direction

• The magnitude \( D \) and direction \( \theta \) of a vector are specified relative to a reference axis (e.g., north of east).

• Angles measured counterclockwise from the positive x-axis are positive; clockwise angles are negative.

• Example: A vector with magnitude 10.3 units and direction 29.1° north of east.

Basic Vector Arithmetic

Operations with Vectors

• Equality: Two vectors are equal if they have the same magnitude and direction, regardless of their initial points.

• Vector Addition: The sum of two vectors is found by placing the tail of the second vector at the head of the first and drawing a vector from the tail of the first to the head of the second.

• Vector Subtraction: Subtracting vector \( \vec{B} \) from \( \vec{A} \) is equivalent to adding \( -\vec{B} \) (the vector with the same magnitude as \( \vec{B} \) but opposite direction) to \( \vec{A} \).

• Negative Vector: The negative of a vector has the same magnitude but the opposite direction.

• Multiplying by a Scalar: When a vector is multiplied by a scalar \( c \):

  • If \( c > 0 \), the direction remains unchanged.

  • If \( c < 0 \), the direction is reversed.

  • The magnitude becomes \( |c| \) times the original.

Example Equation:

Second Newton's Law: $\vec{F}_{\text{net}} = m \vec{a}$

Resolving a Vector into Components

Component Method

• Any vector in a plane can be resolved into two perpendicular components, typically along the x- and y-axes.

• The x-component and y-component are found using trigonometric functions:

$A_x = A \cos \theta$ $A_y = A \sin \theta$

• Where \( \theta \) is the angle the vector makes with the positive x-axis.

• Components can be positive or negative depending on the direction of the vector.

Calculating the Resultant Vector

Pythagorean Theorem and Direction

• The magnitude of the resultant vector \( A \) from its components:

$A = \sqrt{A_x^2 + A_y^2}$

• The direction (angle \( \theta \)) is given by:

$\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$

• The Pythagorean theorem is used for right triangles formed by the components.

Vector Addition and Subtraction Using Components

Component-wise Operations

• For vectors \( \vec{A} \) and \( \vec{B} \):

$C_x = A_x + B_x$ $C_y = A_y + B_y$

• For subtraction:

$D_x = A_x - B_x$ $D_y = A_y - B_y$

• Multiplying by a scalar \( c \):

$E_x = cA_x$ $E_y = cA_y$

Examples and Applications

Worked Examples

• Example 1: Find the x and y components of a velocity vector with magnitude 35.0 m/s at 36.9° west of north.

  • $A_x = A \cos \theta$

  • $A_y = A \sin \theta$

• Example 2: Find the magnitude and direction of a displacement vector with x component -24.0 m and y component -11.0 m.

  • $A = \sqrt{A_x^2 + A_y^2}$

  • $\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$

• Example 3: Vector addition using components for multi-step journeys or operations such as $R = (A + B - 3C)$.

Conceptual and Multiple-Choice Exercises

• Understanding vector orientation, equality, and the effect of operations (e.g., doubling components does not change the angle).

• Recognizing that two vectors are equal only if both magnitude and direction are the same.

• Knowing that if $\vec{A} + \vec{B} = 0$, then $\vec{A}$ and $\vec{B}$ must have the same magnitude but opposite directions.

Summary Table: Scalar vs. Vector Quantities

Quantity Type	Definition	Examples	Mathematical Treatment
Scalar	Magnitude only, no direction	Mass, Temperature, Time	Ordinary algebra
Vector	Magnitude and direction	Displacement, Velocity, Force	Vector algebra (addition, subtraction, components)

Additional info: The notes also include graphical illustrations and real-world examples (e.g., sports, navigation) to contextualize vector operations. The tangent function is used to relate vector direction to its components, and exercises reinforce conceptual understanding.