Lecture Notes on Vector Multiplication and Dot Product

Introduction

• Can vectors be multiplied?
• Yes, with two methods: Dot product and another method (not covered in this lecture)
• Focus on the Dot product today

Dot Product Explanation

• Definition: Dot product adds the products of corresponding vector components.
• Example with vectors ( \textbf{a} ) and ( \textbf{b} ):
  • ( \textbf{a} \cdot \textbf{b} = a_1 \times b_1 + a_2 \times b_2 + a_3 \times b_3 )
• Result: A scalar, not a vector.
• Properties:
  • Commutative: ( \textbf{a} \cdot \textbf{b} = \textbf{b} \cdot \textbf{a} )
  • Distributive over vector addition
  • Associative with scalar multiplication

Practice Example

• Find the dot product of given vectors ( \textbf{v} ) and ( \textbf{w} ).
• Calculate manually to understand the process.

Vector Magnitude and Dot Product

• If ( \textbf{v} \cdot \textbf{v} = |\textbf{v}|^2 )
• Magnitude can be expressed using dot product.

Law of Cosines and Dot Product

• Use Law of Cosines to relate angles and vectors.
• Formula: ( \textbf{v} \cdot \textbf{w} = |\textbf{v}| |\textbf{w}| \cos(\theta) )
  • ( \theta ) is the angle between ( \textbf{v} ) and ( \textbf{w} )
• Application: Find angle between two vectors using dot product.

Identifying Vector Relationships

• Parallel Vectors: Scalar multiples of each other
• Perpendicular Vectors (Orthogonal): Dot product equals zero
• Unit Vectors: Magnitude of 1, used in defining direction

Vector Projection

• Concept: Projecting vector ( \textbf{v} ) onto vector ( \textbf{w} ).
• Vector Projection Formula:
  • ( \text{proj}_\textbf{w} \textbf{v} = \frac{\textbf{v} \cdot \textbf{w}}{|\textbf{w}|^2} \textbf{w} )
• Scalar Projection: Component of ( \textbf{v} ) along ( \textbf{w} )_

Work and Dot Product

• Definition of Work: ( W = |\textbf{F}| |\textbf{d}| \cos(\theta) )
  • ( \textbf{F} ) = force vector, ( \textbf{d} ) = displacement vector
• Relation to Dot Product: ( W = \textbf{F} \cdot \textbf{d} )

Direction Cosines

• Define angles that a vector makes with each axis
• Formulas:
  • ( \cos(\alpha) = \frac{v_1}{|\textbf{v}|} )
  • ( \cos(\beta) = \frac{v_2}{|\textbf{v}|} )
  • ( \cos(\gamma) = \frac{v_3}{|\textbf{v}|} )
• Verify: ( \cos^2(\alpha) + \cos^2(\beta) + \cos^2(\gamma) = 1 )

Conclusion

• Review and practice dot product, vector projection, work, and direction cosines.
• Understand relationships and calculations for application in physics and mathematics.