Lecture Notes: Vector Proofs

Introduction

• Topic: Vector Proofs
• Described as a tough topic, aimed at higher grade levels (grade 8-9 topics).
• Suggestion: Draw diagrams, use pen & paper for practice.

Vector Proof Basics

• Understanding ratios in vectors.
• Example: Point P on line AB such that AP:PB = 3:1.
• Conversion of ratio into fractions: AP = 3/4 of AB, PB = 1/4 of AB.

Finding Vectors

• Task: Find vector OP in terms of vectors A and B.
• Solution involves:
  • Moving from O to P by utilizing known vectors.
  • Options include moving along OA then up to P, or OB then down to P.

Key Steps:

1. Calculate vector AB using path O -> A -> B.
2. Find vector AP: 3/4 of vector AB.
3. Combine OA and AP to get OP.
4. Simplify and factorize the resulting vector.

Proving Collinearity

• Task: Prove points N, M, C are collinear.
• Vector proofs involve finding vectors for sections of the line (e.g., NC, NM, MC).

Key Concepts:

• Use full vectors between points to determine direction (e.g., NC).
• Calculate midpoint vectors (e.g., NM).
• Show that vectors are scalar multiples of each other to prove collinearity.

Example Problems:

• Example 1: Midpoint problem involving vectors NA, NB, and MC.
• Example 2: Prove XYZ is a straight line using vectors. Involves labeling vectors and finding scalar multiples.

Advanced Vector Problem

• Task: Prove ADE is a straight line.
• Convert line proportional problems into vector equations.

Solution Strategy:

1. Identify vectors from A to E and B to C.
2. Use ratios to calculate fractional vectors (e.g., BD, DC).
3. Prove collinearity by showing vectors have the same direction or common vector components.

Tips for Solving Vector Proofs

• Always factorize vectors to find common components.
• Look for lines with no vectors and create vectors for them.
• Practice simplifying and factorizing vector equations.

Conclusion

• Vector proofs are complex and require practice.
• Focus on identifying key lines and calculating vectors correctly.
• Remember to factorize for common directionality.
• Continue practicing with various examples to build familiarity with concepts.