Vectors and 3D Geometry Lecture Notes

Introduction

• Goal: Achieving full marks (80/80) in board exams for Mathematics.
• Focus: Vectors and 3D Geometry.
• Resources: Providing PDFs and solving important board questions.
• Timetable: Following a structured timetable for effective study.
• Telegram Group: Maths की महफिल for doubts and query resolution.

Vectors

Basic Concepts

• Definition: A vector is a quantity that has both direction and magnitude.
• Notations: Initial point (A) and Terminal point (B); written as (\overrightarrow{AB}).
• Formula: (\overrightarrow{AB} = B - A).
• Position Vector: Denoted as (\overrightarrow{OA}), where O is the origin.

Vector Operations

• Addition & Subtraction: (\overrightarrow{A} + \overrightarrow{B} = (a_1 + b_1)\hat{i} + (a_2 + b_2)\hat{j} + (a_3 + b_3)\hat{k}).
• Magnitude: (|\overrightarrow{A}| = \sqrt{x^2 + y^2 + z^2}).
• Unit Vector: (\hat{A} = \frac{\overrightarrow{A}}{|\overrightarrow{A}|}).

Special Vectors

• Zero Vector: Magnitude is zero.
• Position Vector: Represents position in space.
• Unit Vector: Has a magnitude of one.

Vector Products

Dot Product

• Formula: (\overrightarrow{A} \cdot \overrightarrow{B} = |A||B|\cos\theta).
• Properties: Commutative and Distributive.
• Angle Between Vectors: Derived from dot product, if (\theta = 90^\circ), (\overrightarrow{A} \cdot \overrightarrow{B} = 0).

Cross Product

• Formula: (\overrightarrow{A} \times \overrightarrow{B} = |A||B|\sin\theta \hat{n}).
• Properties: Non-Commutative, follows right-hand rule.
• Applications: Finding area of parallelogram and triangle.

Key Equations and Concepts

Section Formula

• Internal Division: (\overrightarrow{C} = \frac{m\overrightarrow{B} + n\overrightarrow{A}}{m+n}).
• Mid-Point Formula: Special case where m=n.

Direction Cosines and Ratios

• Direction Ratios (DRs): Proportional to direction cosines (DCs).
• Direction Cosines (DCs): Cosine values of angles between vector and coordinate axes, summing to one.
• Formula: (\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1).

Straight Lines in 3D

Equations of Lines

• Vector Form: (\overrightarrow{r} = \overrightarrow{a} + \lambda \overrightarrow{b}).
• Cartesian Form: (\frac{x - x_1}{l} = \frac{y - y_1}{m} = \frac{z - z_1}{n}).

Specialized Problems

• Intersection of Lines: Set corresponding equations equal.
• Shortest Distance Between Skew Lines: Using cross product of direction ratios (DRs).
• Parallel and Perpendicular Lines: Check DRs, for perpendicular (\overrightarrow{A} \cdot \overrightarrow{B} = 0).

Plane in 3D

Equation of Plane

• General form: (Ax + By + Cz + D = 0).
• Normal Vector: Perpendicular to the plane.
• Point-Normal Form: (\overrightarrow{r} \cdot \hat{n} = d).

Intersecting and Parallel Planes

• Angle Between Planes: (\cos \theta = \frac{\overrightarrow{n_1} \cdot \overrightarrow{n_2}}{|\overrightarrow{n_1}| |\overrightarrow{n_2}|}).
• Check for parallel planes using normal vectors.

Important Problems and Solutions

Intersection Point for Given Skew Lines

1. Given lines in vector and cartesian forms.
2. Solving to find intersection or shortest distance.
3. Special attention to parameter variations (λ and μ).

Foot of Perpendicular and Image of Point in Line/Plane

1. Concept of mid-point for finding image.
2. Use directional ratios for line equations.
3. Solving step by step for foot of perpendicular.

Distance of Point from Line

1. Shortest distance formula application.
2. Checking perpendicular distances.

Conclusion

• Practice: Extensive practice required for understanding and mastering concepts.
• Resources: Utilize PDFs, Telegram group, and revision lectures.
• Examination Tips: Focus on board-oriented important questions and formulas.

Good luck with your studies and preparation! 🚀