Vectors and Coordinate Systems – Study Notes

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 describing physical phenomena such as motion, forces, and fields.

• Scalar Quantity: A quantity described by a single number (magnitude only), such as mass, temperature, or time.

• Vector Quantity: A quantity that has both magnitude and direction, such as velocity, displacement, force, and acceleration.

• Geometric Representation: Vectors are represented as arrows; the length indicates magnitude, and the arrow points in the direction of the vector.

• Notation: Vectors are denoted with an arrow above the letter, e.g., $\vec{r}$ for position, $\vec{v}$ for velocity, $\vec{a}$ for acceleration.

Properties of Vectors

Vectors can be used to describe displacement, velocity, and other directional quantities. The magnitude and direction are key properties.

• Example: If Sam walks and ends up 200 ft northeast of his starting point, his displacement vector is $\vec{S} = (200\text{ ft}, \text{northeast})$.

• Magnitude: The magnitude of a vector $\vec{S}$ is denoted $|\vec{S}|$ and represents the length of the arrow (distance between initial and final points).

Vector Addition

Vectors can be added graphically or algebraically to find net quantities such as total displacement or resultant 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 $C = \sqrt{A^2 + B^2}$.

• Direction: The direction of the resultant vector can be found using trigonometry, e.g., $\tan \theta = \frac{B}{A}$.

• Example: If a hiker walks 3 miles east and then 4 miles north, the net displacement is $C = \sqrt{3^2 + 4^2} = 5$ miles, at an angle $\theta = \tan^{-1}(4/3)$ north of east.

Graphical and Algebraic Vector Addition

Vectors can be added using graphical methods (drawing arrows) or algebraic methods (adding components).

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

• Algebraic Addition: Add the corresponding components of each vector.

• Example: If $\vec{A} = (100\text{ m}, \text{east})$ and $\vec{B} = (50\text{ m}, \text{northwest})$, decompose $\vec{B}$ into $x$ and $y$ components and add to $\vec{A}$.

Coordinate Systems and Vector Components

A coordinate system is a grid imposed on a problem to define positions and directions. Vectors can be decomposed into components parallel to the axes of the coordinate system.

• Origin: The reference point for the coordinate system.

• Axes Orientation: Axes can be oriented as needed; typically, $x$ is horizontal and $y$ is vertical.

• Component Vectors: Any vector $\vec{A}$ can be written as the sum of its $x$ and $y$ components: $\vec{A} = A_x \hat{i} + A_y \hat{j}$.

• Decomposition: $A_x$ is the projection of $\vec{A}$ onto the $x$-axis, $A_y$ onto the $y$-axis.

Determining Vector Components

To find the components of a vector, use trigonometric relationships based on the vector's magnitude and direction.

• Formulas:

  • $A_x = A \cos \theta$

  • $A_y = A \sin \theta$

• Signs: The sign of each component depends on the direction of the vector relative to the axes.

• Example: A rabbit runs at $10.0\text{ m/s}$, $40.0^\circ$ north of west. $v_x = (10.0\text{ m/s}) \cos 40.0^\circ = -7.66\text{ m/s}$, $v_y = (10.0\text{ m/s}) \sin 40.0^\circ = 6.43\text{ m/s}$.

Unit Vectors

Unit vectors are vectors of length 1 that indicate direction along coordinate axes.

• Symbols: $\hat{i}$ (positive $x$ direction), $\hat{j}$ (positive $y$ direction).

• Usage: Any vector can be expressed as a sum of its components multiplied by unit vectors: $\vec{A} = A_x \hat{i} + A_y \hat{j}$.

Vector Algebra

Vector operations include addition, subtraction, and multiplication by scalars.

• Addition: $\vec{C} = \vec{A} + \vec{B}$, where $C_x = A_x + B_x$, $C_y = A_y + B_y$.

• Subtraction: $\vec{R} = \vec{P} - \vec{Q}$, where $R_x = P_x - Q_x$, $R_y = P_y - Q_y$.

• Scalar Multiplication: $\vec{T} = c\vec{S}$, where $T_x = cS_x$, $T_y = cS_y$.

Summary Table: Scalar vs. Vector Quantities

Quantity Type	Examples	Properties
Scalar	Mass, Temperature, Time	Magnitude only
Vector	Displacement, Velocity, Force	Magnitude and Direction

Additional info:

• Vectors are used in all areas of physics, including kinematics, dynamics, electromagnetism, and fluid mechanics.

• Coordinate systems can be rotated or tilted to simplify problems, especially when analyzing motion along inclined planes or surfaces.