Vectors, Scalars, and Basic Physics Quantities: Study Notes

Math Review and Physics Quantities

Scalars and Vectors

In physics, quantities are classified as either scalars or vectors. Understanding the distinction is fundamental for describing motion and forces.

• Scalar: A quantity with only magnitude (size). Examples: mass, temperature, time, distance.

• Vector: A quantity with both magnitude and direction. Examples: displacement, velocity, acceleration, force.

Quantity	Scalar	Vector
Distance/Displacement	Distance	Displacement
Speed/Velocity	Speed	Velocity
Mass	Mass	-
Force	-	Force
Acceleration	-	Acceleration

Additional info: Scalars are added arithmetically, while vectors require vector addition rules.

SI Units and Base Quantities

Physics uses the International System of Units (SI) for consistency. The main base units are:

• Second (s): Time

• Kilogram (kg): Mass

• Meter (m): Length

• Ampere (A): Electric current

• Kelvin (K): Temperature

• Mole (mol): Amount of substance

• Candela (cd): Luminous intensity

Forces, Momentum, and Energy

Force, Mass, and Acceleration

Newton's Second Law relates force, mass, and acceleration:

• Formula: $F = m a$

• Units: Newton (N), where $1\ \mathrm{N} = 1\ \mathrm{kg} \cdot \mathrm{m}/\mathrm{s}^2$

Work and Energy

Work is done when a force moves an object through a distance:

• Formula: $W = F d$

• Units: Joule (J), where $1\ \mathrm{J} = 1\ \mathrm{N} \cdot \mathrm{m}$

Momentum

Momentum is the product of mass and velocity:

• Formula: $p = m v$

• Units: $\mathrm{kg} \cdot \mathrm{m}/\mathrm{s}$

Vectors: Representation and Operations

Vector Representation

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

• Notation: $\vec{A}$ or boldface A

• Components: Vectors can be broken into horizontal (x) and vertical (y) components.

Vector Addition and Subtraction

Vectors are added or subtracted using graphical or analytical methods.

• Tip-to-Tail Method: Place the tail of the second vector at the tip of the first. The resultant vector is drawn from the tail of the first to the tip of the last.

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

• Subtraction: To subtract $\vec{B}$ from $\vec{A}$, add $\vec{A}$ and $-\vec{B}$ (reverse direction of $\vec{B}$).

Resolving Vectors into Components

Any vector can be resolved into perpendicular components, usually along the x and y axes. This is essential for analyzing motion and forces in two dimensions.

• Horizontal component: $A_x = A \cos \theta$

• Vertical component: $A_y = A \sin \theta$

• Magnitude from components: $A = \sqrt{A_x^2 + A_y^2}$

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

Example: Finding Components

Given a vector of magnitude 20 N at 30° north of east:

• $V_x = 20 \cos 30^\circ = 17.3\ \mathrm{N}$ (horizontal)

• $V_y = 20 \sin 30^\circ = 10\ \mathrm{N}$ (vertical)

Summary Table: Scalars vs. Vectors

Property	Scalar	Vector
Magnitude	Yes	Yes
Direction	No	Yes
Addition Rule	Arithmetic	Vector addition
Examples	Mass, Time, Speed	Displacement, Velocity, Force