Lecture Notes: Exercise 10.3 - Vector Questions

Introduction

• Objective: Find the angle between two vectors A and B with magnitudes √3 and 2, given A⋅B = √6.

Key Concepts

• Vectors: Represented as A⃗ and B⃗.
• Magnitudes: |A| = √3, |B| = 2.
• Dot Product: A⋅B = √6.

Formulas

• Dot Product Formula: A⋅B = |A| |B| cos θ.
• Simplify to find cos θ: cos θ = (A⋅B) / (|A| |B|).

Simplification

• Given A⋅B = √6, rewrite √6 as √3 * √2.
• cos θ = (√3 * √2) / (√3 * 2).
• Simplify: cos θ = 1 / √2.
• cos θ = 1/√2 is true for θ = π/4.
• Angle θ = π/4.

Finding the Angle between Vectors Using Dot Product

• Vectors: A⃗ = i - 2j + 3k, B⃗ = 3i - 2j + k.
• Requirements: A⋅B and magnitudes |A|, |B|.
• Calculate A⋅B: 13 + (-2)(-2) + 3*1 = 3 + 4 + 3 = 10.
• Magnitude of A: √(1^2 + (-2)^2 + 3^2) = √14.
• Magnitude of B: √(3^2 + (-2)^2 + 1^2) = √14.
• cos θ = (A⋅B) / (|A| |B|) = 10 / 14.

Projection of Vector A on B

• A⃗ = i - k, B⃗ = i + j.
• Projection Formula: Projection of A on B = (A⋅B) / |B|.
• Calculate A⋅B: (1*1 + (-1)*1) = 0.
• Magnitude of B: √(1^2 + 1^2) = √2.
• Projection = 0 / √2 = 0.

Projection of Vector - Another Example

• A⃗ = i + 3j + 7k, B⃗ = 7i - j + 8k.
• Calculate A⋅B: (17 + 3(-1) + 7*8).
• Magnitude of B: √(7^2 + (-1)^2 + 8^2).

Projection Details

• Projection Formula: (A⋅B) / |B| = (60 / √114).

Unit Vectors

• Given Vectors are Unit Vectors: A, B, C with |A| = |B| = |C| = 1.
• Check perpendicularity through dot product: a.b = 0.
• B.D: Unit Vectors parallel.

Finding Magnitude

• Magnitude of X, |A| Given, Unit Vector.
• Use formula for finding magnitude with given unit vector properties.

Solution Methodologies

• Finding λ: A + λB ⟂ C
• Derived Values: A + λB, Dot Product with C = 0 to find λ.

Perpendicular Vectors

• Checking Perpendicularity: Using Magnitude and Dot Product.
• Magnitude Comparisons: Simplification of products.

Scalar and Vector Properties

• Different Methods depending on given properties.
• Evaluating magnitudes and dot products.

Real-angle triangles and Vector Properties

• Utilizing right-angle property.
• Distance Methodology: Using subtracting and dot-product methods for vectors.
• Ensuring collinear points through distance measures.

Examples and Applications

• Practical scenarios provided with step-by-step solutions and various methods to ensure understanding.

Important Points

• Double-check dot products and vector magnitudes.
• Right-angle triangles and their properties in vectors.
• Repeated verification of results.

Conclusion

• Recap of necessary vector properties and methodologies.
• Important formulas and concepts for exam preparation.

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• Use these notes as a reference for vector-related questions, providing step-by-step explanations and simplifications method.*

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