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Vectors

Vector intro for linear algebra​

What is a Vector?​

  • A vector has both magnitude (size) and direction.
  • Example:
    • Not a vector: "Moving at 5 miles per hour" (only magnitude).
    • Vector: "Moving at 5 miles per hour east" (magnitude + direction).
    • Speed is a scalar (only magnitude), while velocity is a vector (magnitude + direction).

Visualizing a Vector​

Visualizing Vector

  • A vector can be represented as an arrow:
    • Arrow length = magnitude of the vector.
    • Arrow direction = direction of the vector.
  • Vectors can be moved around if they keep the same magnitude and direction.

Vector Representation in 2D​

  • In two dimensions, a vector is written as (x, y):
    • The first value (x) represents movement in the horizontal direction.
    • The second value (y) represents movement in the vertical direction.
  • Example:
    • A vector v moving 5 units east: (5, 0)
    • A vector a moving 3 right and 4 up: (3, 4)

Column Vector Notation​

  • Instead of writing as a row (x, y), vectors are often written as column vectors:
[ 5 ]
[ 0 ]
  • This format is widely used in linear algebra.

Magnitude of a Vector​

  • The length of a vector can be found using the Pythagorean theorem:
  • For a vector (3, 4): 
    Magnitude = √(3² + 4²) = √9 + 16 = √25 = 5
  • This is useful in determining the size of a vector.

Extending Vectors to Higher Dimensions​

  • Linear algebra allows working with vectors in 3D, 4D, or even higher dimensions.
  • While we can visualize up to 3D, mathematical notation helps work with higher dimensions.

Real coordinate spaces​

What is R² (Two-Dimensional Real Coordinate Space)?​

  • Notation: R² (or ℝ²) represents two-dimensional real coordinate space.
  • Meaning: It includes all possible ordered pairs (x, y) of real numbers.
  • Example:
    • A vector (3,4) belongs to R².
    • A vector (4,3) is different from (3,4) because order matters.
  • Visual Representation:
    • Vectors in R² can be represented as arrows in a 2D coordinate plane.
    • Example: The vector (4,3) moves 4 units right and 3 units up.

What is a Tuple?​

  • A tuple is an ordered list of numbers.
  • A 2-tuple is an ordered pair (x, y).
  • In R², all numbers in the tuple must be real numbers (no imaginary numbers).
  • Example:
    • (3,4) and (-3,-4) are both in R².
    • (i, 2) is not in R² because i is imaginary.

What is R³ (Three-Dimensional Real Coordinate Space)?​

  • Notation: R³ (or ℝ³) represents three-dimensional real coordinate space.
  • Meaning: It includes all possible ordered triplets (x, y, z) of real numbers.
  • Example:
    • A vector (2, -1, 4) belongs to R³.
    • A vector (-1, 5, 3) is also in R³.
  • Visual Representation:
    • Vectors in R³ can be drawn in a 3D coordinate system with x, y, and z axes.

What is Rⁿ (Higher-Dimensional Real Coordinate Space)?​

  • Notation: Rⁿ (or ℝⁿ) represents n-dimensional real coordinate space.
  • Meaning: It includes all possible ordered n-tuples (x₁, x₂, ..., xₙ) of real numbers.
  • Example:
    • A 4D vector (1,2,3,4) is in R⁴.
    • A 100D vector (x₁, x₂, ..., x₁₀₀) is in R¹⁰⁰.
  • Visualization:
    • R³ is easy to visualize.
    • R⁴ and beyond cannot be visualized, but can be represented mathematically.

What is NOT in Rⁿ?​

  • A vector with fewer dimensions is not in a higher-dimensional space:
    • Example: (3,4) is in R², but not in R³.
  • A vector with imaginary numbers is not in Rⁿ:
    • Example: (i, 2, 3) is not in R³ because i is imaginary.

Adding vectors algebraically & graphically​

Adding Vectors Graph

1. Definition of Vector Addition​

  • Given two 2D vectors, a and b, we define their sum by adding corresponding components:
    • If a = (6, -2) and b = (-4, 4):
      • First component: 6 + (-4) = 2
      • Second component: -2 + 4 = 2
    • So, a + b = (2,2).

2. Visual Representation of Vector Addition​

  • Vectors are represented as arrows on the coordinate plane.
  • Magnitude (length of the arrow) and direction define a vector.
  • A vector can be moved if its magnitude and direction remain unchanged.

3. Step-by-Step Visual Addition of Vectors​

  • Draw vector a starting from the origin.
  • Draw vector b starting from the origin (or at the head of vector a).
  • The resultant vector (sum) is the vector that starts at the origin and ends at the tip of the second vector.

4. The "Tip-to-Tail" Method​

  • To add a + b:
    1. Start with vector a from the origin.
    2. Place vector b at the head of vector a.
    3. Draw the resultant vector from the origin to the tip of b.
  • The same works for b + a, proving that vector addition is commutative.

5. Commutative Property of Vector Addition​

  • a + b = b + a:
    • Numerically, adding components in any order gives the same result.
    • Visually, switching the order of the vectors still results in the same resultant vector.

6. Conceptual Understanding​

  • Vector addition represents combined movement or shift in space.
  • It applies to:
    • Displacement (movement in space).
    • Velocity (combining speeds in different directions).
    • Acceleration (combining forces acting on an object).
  • The sum represents the overall effect of two vectors combined.