Overview
This lecture explores two equivalent definitions of the dot product of vectors—one involving angles and magnitudes, and another using vector components—then applies both to find a perpendicular unit vector.
Dot Product Definitions
- The geometric definition: dot product equals product of vector lengths and cosine of the angle between them, i.e. v · w = |v||w|cosθ.
- The component definition: for vectors v=(λ₁, λ₂, λ₃) and w=(μ₁, μ₂, μ₃), v · w = λ₁μ₁ + λ₂μ₂ + λ₃μ₃.
- Both definitions result in a scalar value.
Equivalence Example
- Example vectors: v = (1, 1), w = (0, 3).
- Component method: 1·0 + 1·3 = 3.
- Geometric method: |v| = √2, |w| = 3, angle = 45°, cos(45°)=1/√2; product = √2 × 3 × 1/√2 = 3.
- Both methods yield the same result, illustrating equivalence.
Application: Finding Perpendicular Unit Vector
- Problem: Find vector u = (a, b) of length 1, perpendicular to (3, 7).
- Perpendicularity means (3, 7) · (a, b) = 0; so 3a + 7b = 0.
- Solve for a: a = -7b/3.
- Enforce unit length: √(a² + b²) = 1 → a² + b² = 1.
- Substitute a: (−7b/3)² + b² = 1 → (49/9)b² + b² = 1 → (58/9)b² = 1.
- b² = 9/58 → b = ±√(9/58).
- a = -7/3 × b = -7/3 × ±√(9/58).
- Therefore, perpendicular unit vector u₁ = (−7/3√(9/58), √(9/58)).
- The opposite direction offers u₂ = (7/3√(9/58), −√(9/58)).
Key Terms & Definitions
- Dot Product — Scalar product of two vectors; can be defined with components or using magnitudes and angle between them.
- Perpendicular Vectors — Two vectors are perpendicular (orthogonal) if their dot product is zero.
- Unit Vector — A vector with magnitude (length) of one.
Action Items / Next Steps
- Review textbook proof that both dot product definitions are always equivalent.
- Practice finding unit vectors perpendicular to given vectors using both definitions.