2.1 Vectors in the Plane

Vectors are mathematical tools that pack a punch, combining magnitude and direction into one neat package. They're essential for describing motion, forces, and other physical quantities that have both size and orientation.

Vector operations like addition and multiplication allow us to manipulate these quantities with ease. By breaking vectors down into components and using unit vectors, we can represent complex movements and forces in a simple, standardized way.

Vector Basics

Plane vectors vs scalar quantities

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• Vectors possess both magnitude (size or length) and direction
  • Represented using boldface ($\mathbf{v}$) or arrow notation ($\vec{v}$)
  • Vector magnitude denoted as $|\mathbf{v}|$ or $\|\mathbf{v}\|$
• Scalars only have magnitude without direction
  • Speed, temperature (℃), mass (kg)
• Plane vector examples:
  • Displacement (m), velocity (m/s), acceleration (m/s²), force (N)

Vector operations and calculations

• Vector addition: $\mathbf{v} + \mathbf{w} = (v_1 + w_1, v_2 + w_2)$
  • Graphically, place tail of $\mathbf{w}$ at head of $\mathbf{v}$
• Vector subtraction: $\mathbf{v} - \mathbf{w} = (v_1 - w_1, v_2 - w_2)$
  • Graphically, place tail of $\mathbf{w}$ at head of $\mathbf{v}$, draw resultant from tail of $\mathbf{v}$ to head of $\mathbf{w}$
• Scalar multiplication: $c\mathbf{v} = (cv_1, cv_2)$, $c$ is a scalar
  • Positive scalar changes magnitude, not direction
  • Negative scalar changes both magnitude and direction

Vector Representation

Components and magnitudes of vectors

• Component form: $\mathbf{v} = (v_1, v_2)$, $v_1$ horizontal, $v_2$ vertical components
• Magnitude formula: $|\mathbf{v}| = \sqrt{v_1^2 + v_2^2}$
  • Follows from Pythagorean theorem

Unit vectors and basis notation

• Unit vectors: $\hat{i}$ (horizontal) and $\hat{j}$ (vertical)
  • Magnitude 1, point in positive x and y directions
• Standard basis notation: $\mathbf{v} = v_1\hat{i} + v_2\hat{j}$
  • Expresses vector as linear combination of unit vectors
• Coordinate system: Provides a frame of reference for describing vector positions

Vector Operations and Properties

• Parallelogram law: Method for adding vectors graphically
  • The sum of two vectors forms the diagonal of a parallelogram
• Resultant vector: The vector sum of two or more vectors
• Vector space: A set of vectors that can be added together and multiplied by scalars

Vector Applications

Real-world applications of vectors

• Displacement: change in position
  • Moving 3 units east and 4 units north
• Velocity: rate of change of displacement over time
  • Moving at 5 m/s east and 2 m/s north
• Acceleration: rate of change of velocity over time
  • Accelerating at 2 m/s² east and 1 m/s² south
• Force: push or pull acting on an object
  • 10 N force acting 30° north of east