Vectors and Displacement: Study Notes for Introductory Physics

Vectors and Displacement in Physics

Introduction to Vectors

Vectors are fundamental quantities in physics that possess both magnitude and direction. They are used to represent physical quantities such as displacement, velocity, and force. Understanding how to add, subtract, and resolve vectors is essential for solving problems in mechanics and other areas of physics.

• Vector: A quantity with both magnitude and direction (e.g., displacement, velocity).

• Scalar: A quantity with only magnitude (e.g., mass, temperature).

• Vector Notation: Vectors are often denoted by boldface letters or letters with arrows above them, such as \( \vec{A} \).

Vector Addition and Subtraction

Vectors can be added or subtracted using graphical or analytical methods. The most common analytical method is to resolve vectors into their components along the x and y axes.

• Component Form: Any vector \( \vec{A} \) can be written as \( \vec{A} = A_x \hat{i} + A_y \hat{j} \), where \( A_x \) and \( A_y \) are the components along the x and y axes, respectively.

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

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

• Magnitude of a Vector: \( |\vec{A}| = \sqrt{A_x^2 + A_y^2} \)

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

Applications: Displacement and Path Problems

Displacement is a vector quantity representing the change in position of an object. Problems often involve calculating the resultant displacement after several movements in different directions.

• Example: A hiker walks 1200 m east, then 1500 m at 30° north of east. The total displacement is found by adding the two vectors using components.

• Displacement Vector: \( \vec{D} = \vec{A} + \vec{B} \)

• Component Calculation:

  • For \( \vec{A} \) along x-axis: \( A_x = 1200 \) m, \( A_y = 0 \) m

  • For \( \vec{B} \) at 30°: \( B_x = 1500 \cos(30°) \), \( B_y = 1500 \sin(30°) \)

• Resultant Magnitude: \( |\vec{D}| = \sqrt{(A_x + B_x)^2 + (A_y + B_y)^2} \)

• Resultant Direction: \( \theta = \tan^{-1}\left(\frac{A_y + B_y}{A_x + B_x}\right) \)

Right Triangle and Pythagorean Theorem

Many vector problems involve right triangles, where the Pythagorean theorem is used to relate the sides.

• Pythagorean Theorem: \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse.

• Application: If two sides of a right triangle are known, the third can be found using this theorem.

• Example: If a triangle has sides 7 cm and 11 cm, the third side (if right triangle) is \( \sqrt{7^2 + 11^2} = \sqrt{49 + 121} = \sqrt{170} \approx 13 \) cm.

Vector Direction and Angles

Vectors are often described by their magnitude and the angle they make with a reference axis (usually the positive x-axis).

• Angle Measurement: Angles are measured counterclockwise from the positive x-axis.

• Component Calculation: For a vector \( \vec{A} \) with magnitude \( A \) and angle \( \theta \):

  • \( A_x = A \cos(\theta) \)

  • \( A_y = A \sin(\theta) \)

• Example: A vector of 35 units at 325°: \( A_x = 35 \cos(325°) \), \( A_y = 35 \sin(325°) \)

Displacement Vector in Two Dimensions

When an object moves in two or more directions, its total displacement is the vector sum of each segment.

• Example: A person walks 2.0 km north, then 4.0 km south-east. The displacement vector is found by resolving each segment into components and adding.

• General Formula: \( \vec{D}_{\text{total}} = \sum \vec{D}_i \)

Worked Example: Vector Addition with Angles

Given two vectors with magnitudes and directions, their sum and difference can be found using components.

• Given: \( |\vec{A}| = 35.0 \) units at 325°, \( |\vec{B}| = 25.0 \) units at 120°.

• Step 1: Find components:

  • \( A_x = 35.0 \cos(325°) \)

  • \( A_y = 35.0 \sin(325°) \)

  • \( B_x = 25.0 \cos(120°) \)

  • \( B_y = 25.0 \sin(120°) \)

• Step 2: Add or subtract components for \( \vec{A} + \vec{B} \) or \( \vec{A} - \vec{B} \).

• Step 3: Find magnitude and direction of the resultant.

Classification Table: Scalar vs. Vector Quantities

Quantity	Scalar	Vector
Displacement		✓
Velocity		✓
Acceleration		✓
Distance	✓
Speed	✓

Summary of Key Equations

• Vector components: $A_x = A \cos(\theta)$ $A_y = A \sin(\theta)$

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

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

• Pythagorean theorem: $a^2 + b^2 = c^2$

Example Problem

• Problem: A hiker walks 1200 m east, then 1500 m at 30° north of east. Find the total displacement and direction.

• Solution:

  • \( A_x = 1200 \) m, \( A_y = 0 \) m

  • \( B_x = 1500 \cos(30°) = 1299 \) m

  • \( B_y = 1500 \sin(30°) = 750 \) m

  • Total x: \( 1200 + 1299 = 2499 \) m

  • Total y: \( 0 + 750 = 750 \) m

  • Magnitude: \( \sqrt{2499^2 + 750^2} \approx 2609 \) m

  • Direction: \( \tan^{-1}(750/2499) \approx 16.6° \) north of east

Additional info: These notes expand upon the original questions by providing definitions, formulas, and worked examples relevant to introductory vector and displacement problems in physics.