Vectors and Motion Along a Straight Line: Study Notes for College Physics

Vectors in Physics

Vector Basics

Vectors are fundamental quantities in physics, characterized by both magnitude and direction. They are used to represent physical quantities such as displacement, velocity, and force.

• Definition: A vector is a quantity with both magnitude and direction, often represented as an arrow in space.

• Examples: Displacement, velocity, acceleration, and force are all vector quantities.

• Notation: Vectors are typically denoted by boldface letters (e.g., A, B) or with an arrow above the letter (e.g., \vec{A}).

Vector Operations

Two important operations with vectors are the scalar (dot) product and the vector (cross) product.

• Scalar (Dot) Product: The dot product of two vectors A and B is defined as: $\vec{A} \cdot \vec{B} = |A||B|\cos\theta$ where $\theta$ is the angle between the vectors.

• Maximum Magnitude of Dot Product: The dot product is maximized when $\cos\theta$ is maximized, i.e., when $\theta = 0^\circ$ or $\theta = 180^\circ$ (vectors are parallel or anti-parallel).

• Vector (Cross) Product: The cross product of two vectors A and B is defined as: $\vec{A} \times \vec{B} = |A||B|\sin\theta\ \hat{n}$ where $\hat{n}$ is a unit vector perpendicular to the plane containing A and B, and $\theta$ is the angle between them.

• Maximum Magnitude of Cross Product: The cross product is maximized when $\sin\theta$ is maximized, i.e., when $\theta = 90^\circ$ (vectors are perpendicular).

Application: Vectors in the xy-plane

Consider two vectors A and B in the xy-plane:

• Vector A: In the +x direction.

• Vector B: At an angle $\theta$ from the +x-axis toward the +y-axis.

• Dot Product Maximum: Occurs at $\theta = 0^\circ$ or $\theta = 180^\circ$.

• Cross Product Maximum: Occurs at $\theta = 90^\circ$.

• Magnitudes:

  • Dot product magnitude: $|A||B|$ at $\theta = 0^\circ$ or $180^\circ$.

  • Cross product magnitude: $|A||B|$ at $\theta = 90^\circ$.

Motion Along a Straight Line

Displacement, Time, and Average Velocity

Motion along a straight line is a foundational concept in kinematics, describing how objects move in one dimension.

• Displacement (\Delta x): The change in position of an object along a straight line. $\Delta x = x_{final} - x_{initial}$

• Time Interval (\Delta t): The duration over which the displacement occurs. $\Delta t = t_{final} - t_{initial}$

• Average Velocity (v_{avg}): The rate of change of displacement with respect to time. $v_{avg} = \frac{\Delta x}{\Delta t}$

• Example: If a car moves from $x = 0$ m to $x = 100$ m in $5$ seconds, its average velocity is $v_{avg} = \frac{100\ \text{m}}{5\ \text{s}} = 20\ \text{m/s}$.

Instantaneous Velocity

Instantaneous velocity describes the velocity of an object at a specific moment in time.

• Definition: The instantaneous velocity is the derivative of position with respect to time. $v = \frac{dx}{dt}$

• Physical Meaning: It represents how fast and in what direction an object is moving at a particular instant.

• Example: If $x(t) = 5t^2$, then $v = \frac{d}{dt}(5t^2) = 10t$.

Summary Table: Scalar and Vector Products

Operation	Formula	Maximum Value Condition	Physical Meaning
Dot Product (Scalar)	$\vec{A} \cdot \vec{B} = |A||B|\cos\theta$	$\theta = 0^\circ$ or $180^\circ$	Projection of one vector onto another
Cross Product (Vector)	$\vec{A} \times \vec{B} = |A||B|\sin\theta\ \hat{n}$	$\theta = 90^\circ$	Area of parallelogram formed by vectors; direction given by right-hand rule

Additional info:

• Some context and definitions have been inferred based on standard introductory physics curriculum.

• Specific values and examples have been added for clarity and completeness.