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Scalars, Vectors, and Vector Operations

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Scalars, Vectors, and Vector Operations

Why It Matters

Many physical quantities in later topics are vectors. Direction must be handled correctly for displacement, velocity, acceleration, force, momentum, and fields.

Definition

A scalar has magnitude only. A vector has magnitude and direction.

Key Representations

Vector with unit direction:

Vector addition:

Vector subtraction:

Resolution into components:

Scalars vs Vectors

Examples of scalars include mass, time, distance, speed, energy, temperature, and electric potential.

Examples of vectors include displacement, velocity, acceleration, force, momentum, gravitational field strength, and electric field strength.

Vector Representation

Vectors may be represented using arrows, signs in one dimension, bearings, unit vectors, or components. The magnitude of a vector is a scalar.

Addition, Subtraction, and Resultants

Vectors are added tip-to-tail or by resolving into components. The resultant vector is the single vector equivalent to two or more vectors combined.

Vector subtraction is addition of the negative vector. The vector has the same magnitude as but opposite direction.

Vector Resolution

Vector resolution replaces a vector with perpendicular components that have the same combined effect. The angle must be measured from the axis used in the formula.

This note is a measurement-level bridge. The fuller foundation treatment, including diagrams and deeper applications, is in Vectors.

Common Exam Points

  • Do not add vector magnitudes unless the directions justify it.
  • Resultant magnitude alone is incomplete; direction must also be stated.
  • In one-dimensional motion, signs represent vector direction.
  • For component formulas, define the angle clearly.
  • Unit vectors indicate direction without changing magnitude.

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