CH. 1 vectors

Chapter: Vectors

Introduction to Scalars and Vectors

In physics, quantities are classified as either scalars or vectors. Understanding the distinction between these types is fundamental for solving problems involving motion, force, and other physical phenomena.

• Scalar: A quantity that is completely specified by a single value (magnitude) and has no direction.

• Vector: A quantity that is specified by both a magnitude and a direction.

Examples of scalars include temperature, mass, and time. These quantities can be added, subtracted, multiplied, or divided using the rules of ordinary algebra.

• Temperature: Indicates the degree of hotness or coldness and is measured in degrees Celsius, Fahrenheit, or Kelvin.

• Mass: The amount of matter in an object, measured in kilograms or grams.

• Time: The ongoing sequence of events, measured in seconds, minutes, or hours.

Scalars can be positive or negative, depending on the context (e.g., temperature can be below zero).

Vectors require both magnitude and direction for complete specification. Examples include displacement, velocity, and force.

• Displacement: The change in position of an object, described by both length and direction.

• Velocity: The rate of change of displacement, indicating both speed and direction.

• Force: An interaction that changes the motion of an object, specified by magnitude and direction.

Vector quantities are manipulated using vector algebra, which differs from ordinary algebra due to the inclusion of direction.

Components in Calculations

Vectors can be resolved into components to simplify calculations, especially in two or three dimensions. The components of a vector are its projections along the coordinate axes (usually x and y).

• Component: The part of a vector along a specific axis.

• For a vector A at an angle $\theta$ to the x-axis:

  • $A_x = A \cos \theta$

  • $A_y = A \sin \theta$

• These components allow vectors to be added, subtracted, and multiplied more easily.

Example: If a vector has a magnitude of 10 units and points at 30° above the x-axis:

• $A_x = 10 \cos 30^\circ = 8.66$ units

• $A_y = 10 \sin 30^\circ = 5.00$ units

Components are essential for solving problems involving motion, forces, and other vector quantities in physics.

Additional info: The images provided (athletes jumping, weights, thermometer, clock) illustrate the difference between scalar and vector quantities in real-world contexts. For example, the direction of a jump (vector) versus the weight of an object (scalar).