Chapter 3: Vectors and Coordinate Systems

Introduction to Vectors in Physics

Vectors are fundamental mathematical objects used throughout physics and engineering to represent quantities that have both magnitude and direction. Understanding vectors and their properties is essential for analyzing physical phenomena such as motion, forces, and fields.

• Vectors appear everywhere in physics, from velocities to electric fields and forces.

• The tools and techniques for working with vectors are foundational for further studies and professional applications.

Scalar and Vector Quantities

Physical quantities can be classified as either scalars or vectors, depending on whether they possess direction in addition to magnitude.

• Scalar Quantity: A quantity fully described by a single number (magnitude) and no direction. Examples: mass, temperature, volume, time.

• Vector Quantity: A quantity described by both magnitude and direction. Examples: velocity, displacement, force.

• Geometric Representation: Vectors are represented as arrows, with the length indicating magnitude and the arrowhead indicating direction.

• Notation: Vectors are denoted with an arrow above the letter, e.g., for position, for velocity, for acceleration.

Properties of Vectors

Vectors have specific properties that distinguish them from scalars, including how they are measured and represented.

• Magnitude: The length of the vector arrow, representing the size of the quantity.

• Direction: The orientation of the arrow, indicating the direction of the quantity.

• Example: If Sam walks and ends up 200 ft northeast of his starting point, his displacement vector is .

• The magnitude is , which is the straight-line distance between initial and final positions.

Vector Addition

Vectors can be added together to find resultant quantities, such as net displacement or total force.

• Tip-to-Tail Rule: 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.

• Pythagorean Theorem: For vectors at right angles, the magnitude of the resultant is:

• Direction: The angle of the resultant can be found using trigonometry:

• Example: A hiker walks 3 miles east and then 4 miles north. The net displacement is miles at north of east.

Graphical and Algebraic Vector Addition

Vectors can be added graphically (using arrows) or algebraically (using components).

• Graphical Addition: Use the tip-to-tail or parallelogram method to find the resultant vector.

• Algebraic Addition: Add corresponding components:

If and , then

• Law of Cosines: For non-right-angle vectors:

• Law of Sines: Used to find angles in vector addition problems.

Component Vectors and Coordinate Systems

Vectors can be decomposed into components along chosen coordinate axes, simplifying calculations and analysis.

• Coordinate System: An artificial grid imposed on a problem, with an origin and axes (usually x and y).

• Component Vectors: Any vector can be broken into (parallel to x-axis) and (parallel to y-axis).

• Decomposition: , where and are unit vectors in the x and y directions, respectively.

• Magnitude of Components: ,

• Sign Convention: The sign of each component depends on the direction relative to the axes.

Unit Vectors

Unit vectors are vectors of magnitude 1, used to specify direction along coordinate axes.

• : Unit vector in the positive x-direction.

• : Unit vector in the positive y-direction.

• Any vector can be written as:

Worked Example: Rabbit's Velocity

A rabbit runs at 10.0 m/s, 40.0° north of west. Find its velocity components.

• Let positive x-axis point east, positive y-axis point north.

• Velocity vector:

• Components:

• Velocity vector:

Summary Table: Scalar vs. Vector Quantities

Key Equations

• Magnitude of a vector:

• Direction (angle):

• Vector addition (components):

• Decomposition: ,

Applications and Importance

• Vectors are used to describe motion, forces, and fields in physics.

• Understanding vector addition and decomposition is crucial for solving problems in mechanics, electromagnetism, and fluid dynamics.

Additional info: These notes expand on the brief points in the slides, providing full definitions, examples, and equations for clarity and completeness.