1. Understand the Syllabus and Weightage

• Key Topics to Cover:
  1. Basics of Vectors: Magnitude and direction
  2. Types of Vectors: Unit vector, Zero vector, Position vector
  3. Addition and Subtraction of Vectors
  4. Scalar (Dot) Product and its properties
  5. Vector (Cross) Product and its properties
  6. Applications: Angle between two vectors, Projection of a vector
  7. Scalar Triple Product (STP)
• Weightage in JEE Mains:
  • Vector Algebra carries 2-3 questions, equivalent to 8-12 marks.

2. Step-by-Step Study Plan

Step 1: Learn the Basics (Day 1-3)

• Concepts to Master:
  • Magnitude, Unit vectors, Position vectors
  • Types of vectors (Collinear, Coplanar, etc.)
  • Vector addition and subtraction
• Practice:
  • Solve basic problems from NCERT and RD Sharma to build a foundation.
  • Understand graphical representation of vector addition and parallelogram law.

Step 2: Master Dot Product (Day 4-6)

• Concepts:
  • Formula: $\mathbf{a}\cdot \mathbf{b}=|a||b|\cos \theta$
  • Angle between two vectors
  • Properties: Commutative, Distributive
  • Applications: Work done by force (in physics)
• Practice:
  • Solve conceptual questions on angles and orthogonal vectors.
  • Use NCERT Exemplar or HC Verma (Physics) for application-based problems.

Step 3: Master Cross Product (Day 7-9)

• Concepts:
  • Formula: a×b=∣a∣∣b∣sin⁡θ n^\mathbf{a} \times \mathbf{b} = |a||b| \sin\theta \, \hat{n}a×b=∣a∣∣b∣sinθn^
  • Right-hand rule for direction of cross product
  • Properties: Anti-commutative, Distributive
  • Applications: Area of parallelogram and triangle
• Practice:
  • Solve numerical problems on vector products from Arihant or Cengage.
  • Try problems where unit vector components are used.

Step 4: Learn Scalar Triple Product (STP) (Day 10-12)

• Concepts:
  • Formula: $\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})$
  • Applications: Volume of parallelepiped, coplanarity condition.
  • STP is zero if vectors are coplanar.
• Practice:
  • Solve problems involving volume and coplanarity conditions.
  • Use JEE Advanced Previous Papers for STP-related questions.

Step 5: Solve Previous Year Questions (Day 13-15)

• Importance:
  • This will give you a clear idea of the pattern of questions asked in JEE Mains.
• Sources:
  • Use Arihant’s 40 Years JEE Mains & Advanced book or Embibe for topic-wise PYQs.

Step 6: Take Mock Tests & Sectional Tests (Weekly)

• Take sectional tests to get accustomed to time management.
• Analyze your mistakes and note down tricky formulas or concepts.
• Use NTA Abhyas App or Allen mock tests to simulate real-exam scenarios.

3. Study Materials and Books

• NCERT Mathematics (Class 11 and 12) – For conceptual clarity
• Arihant Skills in Mathematics – Vectors & 3D Geometry
• Cengage Mathematics – For practice and advanced questions
• Previous Year JEE Papers – Practice regularly to build confidence

4. Revision and Formula Sheet

• Prepare a formula sheet with all important properties:
  • Dot Product: $\mathbf{a}\cdot \mathbf{b}=|a||b|\cos \theta$
  • Cross Product: a×b=∣a∣∣b∣sin⁡θ n^\mathbf{a} \times \mathbf{b} = |a||b| \sin\theta \, \hat{n}a×b=∣a∣∣b∣sinθn^
  • STP: $\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})$ = Volume of parallelepiped.
• Quick Tip: Revise this formula sheet regularly, especially in the last month before the exam.

5. Key Tips for Success

1. Visualize concepts: Graphically understand vector operations like addition, cross product, etc.
2. Practice consistently: Don’t skip daily practice—15-20 vector problems per day will suffice.
3. Use error logs: Keep track of mistakes made during tests and correct them.
4. Focus on conceptual clarity: Avoid rote learning; understand the why behind formulas.
5. Stay motivated and consistent: Set small goals and track progress weekly.

By following this plan and practicing regularly, you can easily master Vector Algebra and secure marks in IIT JEE Mains 2025. Let me know if you need any further help!