Lecture Notes: Dot Product of Vectors

Introduction

• Topic: Dot Product (also known as Scalar Product)
• Purpose: To define and describe the dot product operation on vectors
• Key Concept: Dot product results in a scalar

Definition of Dot Product

• Vectors Involved: Two vectors, ( u ) and ( v )
• Angle Consideration: ( \theta ), the angle between vectors ( u ) and ( v )
• Mathematical Definition: ( u \cdot v = |u| |v| \cos(\theta) )
  • ( |u| ) = magnitude of ( u )
  • ( |v| ) = magnitude of ( v )
  • ( \cos(\theta) ) = cosine of angle ( \theta )

Conceptual Understanding

• Interpretation: Dot product measures how much two vectors pull in the same direction
• Example 1: Basis Vectors
  • Vectors: ( j ) and ( k ) (unit vectors)
  • Dot Product Calculation:
    • Magnitude of ( j ) = 1, Magnitude of ( k ) = 1
    • Angle between ( j ) and ( k ) = 90 degrees
    • ( j \cdot k = 1 \times 1 \times \cos 90^\circ = 0 )
  • Conclusion: ( j ) and ( k ) are perpendicular, hence no pulling together

Vector Dot Product with Itself

• Vector: ( v )
• Dot Product Calculation:
  • ( v \cdot v = |v|^2 \cos 0 = |v|^2 )
  • Conclusion: Dot product with itself gives the magnitude squared

Dot Product with Opposite Vector

• Vectors: ( v ) and ( -v )
• Dot Product Calculation:
  • Magnitude of ( v ) = Magnitude of ( -v )
  • Angle between ( v ) and ( -v ) = 180 degrees
  • ( v \cdot -v = |v|^2 \cos 180^\circ = -|v|^2 )
  • Conclusion: Opposite direction results in a negative product

Perpendicular Vectors

• Rule: Dot product of perpendicular vectors equals 0
• Reasoning: Cosine of 90 degrees is 0

Dot Product in Coordinates

• Vectors: ( u = ai + bj + ck ), ( v = xi + yj + zk )
• Dot Product Rule:
  • ( u \cdot v = ax + by + cz )
  • Multiply corresponding components and sum them

Example Calculation

• Given Vectors: ( 2i - j + 3k ) and ( 4i + 2j - k )
• Dot Product Calculation:
  • ( 2 \times 4 + (-1) \times 2 + 3 \times (-1) = 8 - 2 - 3 = 3 )
• Conclusion: Dot product using coordinates is straightforward