Calculus of Several Variables - Lecture 2: Inner Product and Distance

Jun 21, 2024

Calculus of Several Variables - Lecture 2

Title: Inner Product and Distance

Key Concepts

  1. Inner Product
    • Fundamental role in calculus of several variables.
    • Explained for 2-dimensional vectors initially.
      • Plane: x-axis (horizontal) and y-axis (vertical).
      • Points: P(x1, y1) and Q(x2, y2) with origin O(0,0).
      • Vectors: OP (R1) and OQ (R2).
      • Representation in row or column vectors for simplicity.
  2. Definition of Inner Product (Dot Product) for 2D Vectors
    • Inner Product of vectors R1 and R2:
      • Formula: ( R1 \cdot R2 = x1 \cdot x2 + y1 \cdot y2 )
  3. Extending Inner Product to n dimensions
    • Vectors in n dimensions: R1 = (x1, x2,...,xn), R2 = (y1, y2,...,yn).
    • Formula: ( R1 \cdot R2 = \sum_{i=1}^{n} x_i y_i ) or ( R1 \cdot R2 = X_{\mu} Y_{\mu} ) (using Einstein summation convention).
    • Holds more algebraic significance in higher dimensions.
  4. Distance in 3D Space
    • Vectors in 3D: OP represented as (x, y, z).
    • Distance (norm) calculation:
      • Formula: ( ||OP|| = \sqrt{x^2 + y^2 + z^2} ) (using Pythagoras theorem).
    • Link to Inner Product: ( ||R|| ^2 = R \cdot R ), where R is position vector OP.
  5. Properties of Vector Length (Norm)
    • Non-negativity: ( ||R|| \geq 0 ).
    • Zero Length: ( ||R|| = 0 ) iff R = 0 vector.
    • Geometric Representation: vectors position points in space, maintains direction and magnitude.
  6. Dot Product with Angle between Vectors
    • Formula: ( A \cdot B = ||A|| ||B|| \cos{\theta} ).
      • ( \theta ) is the angle between vectors A and B.
      • ( \theta = \cos^{-1}{\left(\frac{A \cdot B}{||A||||B||}\right)} ).
      • If ( A \cdot B = 0 ), then vectors A and B are perpendicular.
  7. Cauchy-Schwarz Inequality
    • Statement: ( |A \cdot B| \leq ||A|| ||B|| ).
    • Proof: Uses cosine law and properties of dot product.
  8. Triangle Inequality
    • Statement: ( ||A + B|| \leq ||A|| + ||B|| ).
    • Proof involves squaring norms and applying Cauchy-Schwarz inequality.
    • Useful in proving geometric principles in vectors.

Additional Topics

  • Projection Formula: Details in upcoming assignment.
  • Notes and Additional Material: To be provided by course staff.

Instructor's Remarks

  • Interaction and queries are encouraged for better understanding.
  • Essential to discuss and clarify doubts in mathematics.

[Music] [Music] [Music]