Vectors, Scalars, and Vector Components: Foundations for Physics with Calculus

Vectors vs. Scalars

Definitions and Key Differences

In physics, quantities are classified as either scalars or vectors. Understanding this distinction is fundamental for describing physical phenomena accurately.

• Scalar Quantity: A physical quantity that is described completely by its magnitude (size or amount) alone. It has no associated direction.

• Vector Quantity: A physical quantity that requires both magnitude and direction for a complete description.

Examples:

Scalar Quantities	Vector Quantities
Time	Displacement
Temperature	Velocity
Distance traveled	Acceleration
Speed	Momentum
Mass
Energy
Power

• Magnitude: The size or strength of a quantity, always positive for vectors.

• Direction: The orientation in space of a vector quantity.

Example: Speed (scalar) tells you how fast you are moving, while velocity (vector) tells you how fast and in what direction.

Vector Notation and Representation

How Vectors Are Written and Interpreted

Vectors are typically represented in boldface (e.g., A) or with an arrow overhead (e.g., $\vec{A}$). The magnitude is written as $A$ (italicized, always positive), and the direction is specified by an angle $\theta$ or other conventions.

• Full Vector: $\vec{A}$ or A (includes both magnitude and direction)

• Magnitude Only: $A = |\vec{A}|$

• Direction: $\theta_A$ (measured from a reference axis, usually the positive x-axis, counterclockwise is positive)

Other ways to specify direction include compass bearings (measured from North) and descriptive phrases (e.g., "degrees above the horizontal").

Example: The initial velocity of a soccer ball: $\vec{v}_0 = 35\ \mathrm{m/s}$ at $40^\circ$ above the horizontal.

Vector Components

Breaking Vectors into Perpendicular Parts

Any vector in a plane can be expressed as the sum of its components along the x- and y-axes. This is essential for analyzing motion and forces in physics.

• Component Form (2D): $\vec{A} = (A_x, A_y)$ or $\vec{A} = A_x \hat{i} + A_y \hat{j}$

• Component Form (3D): $\vec{A} = (A_x, A_y, A_z)$ or $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$

• Unit Vectors: $\hat{i}$ (x-direction), $\hat{j}$ (y-direction), $\hat{k}$ (z-direction)

Finding Components:

• $A_x = A \cos \theta$

• $A_y = A \sin \theta$

Here, $A$ is the magnitude and $\theta$ is the angle measured from the positive x-axis.

Example: For $\vec{v}_0 = 35\ \mathrm{m/s}$ at $40^\circ$ above the horizontal:

• $v_{0x} = 35 \cos 40^\circ \approx 26.8\ \mathrm{m/s}$

• $v_{0y} = 35 \sin 40^\circ \approx 22.5\ \mathrm{m/s}$

Converting Between Magnitude/Direction and Components

Using Trigonometry to Switch Forms

Given the components $A_x$ and $A_y$, the magnitude and direction of the vector can be found using:

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

• Direction: $\theta = \arctan\left(\frac{A_y}{A_x}\right)$ (be careful with the quadrant!)

When using inverse trigonometric functions, always check which quadrant the vector lies in to determine the correct angle.

Example: If $A_x = -0.79\ \mathrm{m}$ and $A_y = -2.16\ \mathrm{m}$, then:

• $A = \sqrt{(-0.79)^2 + (-2.16)^2} \approx 2.3\ \mathrm{m}$

• $\theta = \arctan\left(\frac{-2.16}{-0.79}\right) \approx 250^\circ$ (after adjusting for the correct quadrant)

Review: Properties of Vectors and Scalars

Quick Facts and Common Questions

• The magnitude of a vector is always positive and can have units.

• Vector components can be positive or negative and have the same units as the vector.

• Scalars never have direction; vectors always do.

Example Table: Classification of Quantities

Quantity	Scalar or Vector?
Time	Scalar
Temperature	Scalar
Displacement	Vector
Distance traveled	Scalar
Velocity	Vector
Speed	Scalar
Acceleration	Vector
Mass	Scalar
Energy	Scalar
Power	Scalar
Momentum	Vector

Summary Table: Scalar vs. Vector Quantities

Scalar Quantities	Vector Quantities
Time	Displacement
Temperature	Velocity
Distance traveled	Acceleration
Speed	Momentum
Mass
Energy
Power