Chapter 3: Vectors and Coordinate Systems – Study Notes

Vectors and Coordinate Systems

Introduction to Vectors

In physics, quantities are classified as either scalars or vectors. Scalars are described by a single number (magnitude) and units, such as mass or temperature. Vectors, on the other hand, have both magnitude and direction, and are essential for describing motion and forces in physics.

• Scalar Quantity: Defined by magnitude only (e.g., mass, temperature, volume).

• Vector Quantity: Defined by both magnitude and direction (e.g., displacement, velocity, acceleration).

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

• Notation: Vectors are denoted with an arrow above the letter, such as $\vec{v}$ for velocity.

Properties of Vectors

Vectors are independent of their initial position; only their magnitude and direction matter. Two vectors are equal if they have the same magnitude and direction, regardless of their starting points.

• Displacement Example: If two people walk 200 ft northeast from different starting points, their displacement vectors are equal.

• Equality of Vectors: $\vec{A} = \vec{B}$ if $|\vec{A}| = |\vec{B}|$ and both point in the same direction.

Vector Addition

Vectors can be added graphically or algebraically. The tip-to-tail method and the parallelogram rule are common graphical techniques. When vectors are at right angles, the Pythagorean theorem is used to find the resultant's magnitude.

• 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 Rule: Place both vectors with their tails together; the diagonal of the parallelogram represents the sum.

• Pythagorean Theorem (for perpendicular vectors): $$ |\vec{R}| = \sqrt{A^2 + B^2}$$

• Direction (angle θ): $$ \theta = \tan^{-1}\left(\frac{B}{A}\right)$$

Example: A hiker walks 4 miles east, then 3 miles north. The net displacement is $\sqrt{4^2 + 3^2} = 5$ miles at $\tan^{-1}(3/4) = 36.9^\circ$ north of east.

Addition of More Than Two Vectors

Vector addition extends to any number of vectors by repeated application of the tip-to-tail method or by adding all components algebraically.

• Net Displacement: The sum of all displacement vectors gives the total change in position.

• Algebraic Addition: Add all x-components and all y-components separately.

Coordinate Systems and Vector Components

A coordinate system is a grid used to specify positions and directions. The most common is the Cartesian (x, y) system, divided into four quadrants. The choice of origin and axis orientation is arbitrary and should suit the problem.

• Component Vectors: Any vector can be decomposed into two perpendicular components along the axes.

• Decomposition: $\vec{A} = \vec{A}_x + \vec{A}_y$

• Component: The signed magnitude of the projection of the vector onto an axis.

Determining Vector Components

To find the components of a vector $\vec{A}$ with magnitude $A$ and angle $\theta$ (measured from the x-axis):

• x-component: $$ A_x = A \cos \theta$$

• y-component: $$ A_y = A \sin \theta$$

• If the vector points left or down, insert a negative sign as appropriate.

Example: A vector of 5 units at 30° above the x-axis has components $A_x = 5 \cos 30^\circ = 4.33$, $A_y = 5 \sin 30^\circ = 2.5$.

Moving Between Geometric and Component Representations

Vectors can be described by their magnitude and direction (geometric) or by their components (algebraic). To convert between these:

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

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

Note: Always check the signs of components to determine the correct quadrant for the angle.

Unit Vectors

Unit vectors have magnitude 1 and indicate direction along coordinate axes. In two dimensions:

• î: Unit vector in the +x direction

• ĵ: Unit vector in the +y direction

• Any vector can be written as $\vec{A} = A_x \hat{i} + A_y \hat{j}$

Vector Algebra

Vectors can be added, subtracted, and multiplied by scalars using their components:

• Addition: $\vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$

• Subtraction: $\vec{A} - \vec{B} = (A_x - B_x)\hat{i} + (A_y - B_y)\hat{j}$

• Scalar Multiplication: $c\vec{A} = (cA_x)\hat{i} + (cA_y)\hat{j}$

Tilted Axes and Arbitrary Directions

Sometimes, it is convenient to tilt the coordinate axes to align with a surface or direction of interest. The axes remain perpendicular, but are not necessarily horizontal or vertical.

• Application: Useful for analyzing forces or motion parallel and perpendicular to an inclined plane or bone.

• Component Calculation: Use the same trigonometric methods, but with respect to the new axes.

Example: A muscle pulls at 15° to a bone; the force components parallel and perpendicular to the bone are found using $F_{\parallel} = F \cos 15^\circ$, $F_{\perp} = F \sin 15^\circ$.

Summary Table: Vector Operations

Operation	Equation	Description
Magnitude from components	$A = \sqrt{A_x^2 + A_y^2}$	Finds the length of the vector
Direction from components	$\theta = \tan^{-1}(A_y/A_x)$	Finds the angle with respect to x-axis
Component along x	$A_x = A \cos \theta$	Projection onto x-axis
Component along y	$A_y = A \sin \theta$	Projection onto y-axis
Vector addition	$\vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$	Adds corresponding components
Scalar multiplication	$c\vec{A} = (cA_x)\hat{i} + (cA_y)\hat{j}$	Multiplies each component by scalar c

Key Takeaways

• Vectors are essential for describing physical quantities with direction.

• Vector addition can be performed graphically or algebraically.

• Decomposing vectors into components simplifies calculations.

• Unit vectors provide a concise way to express vector components.

• Coordinate systems can be chosen or tilted to suit the problem.

Additional info: In practice, always check the direction and sign of each component, especially when working in non-standard or tilted coordinate systems. Mastery of vector operations is foundational for all subsequent topics in physics, including kinematics, dynamics, and electromagnetism.