Physics & MathematicsNCTB Physics Paper 1 Chapter 2 (Vectors)NCERT Class 11 VectorsBUET & Engineering Admission
3D Vector Math & Geometric Solver
Online 2D and 3D vector calculator: dot product, cross product, angle between vectors, unit vectors, projection, and parallelogram area.
3D Vector Math & Geometric Solver
Calculate dot product, cross product, angle, unit vectors, projection, and area in real time.
Quick Presets:
🔵 Vector A|A| = 3.7417
🟣 Vector B|B| = 4.5826
2D XY Plane Vector Diagram
— Vector A— Vector B-- A+B
স্কেলার গুণন (A · B)3ax·bx + ay·by + az·bz
মধ্যবর্তী কোণ (θ)79.92°1.3949 rad
লম্ব শর্ত (A · B = 0)Not Orthogonal
সমান্তরাল শর্তNot Parallel
সামান্তরিকের ক্ষেত্রফল16.8819|A × B| sq units
ত্রিভুজের ক্ষেত্রফল8.441½ |A × B| sq units
📝 Step-by-Step Mathematical Evaluation:
ভেক্টর যোগফল (A + B):
ভেক্টর বিয়োগফল (A - B):
ভেক্টর গুণন বা ক্রস প্রোডাক্ট (A × B):
B-এর ওপর A-এর স্কেলার অভিক্ষেপ:
Formula & Derivation
Theoretical foundation, dimensional analysis, and governing boundary conditions.
Solved Textbook Examples
Three fully worked pedagogical exemplars: standard textbook, advanced edge-case, and authentic past board examination.
Higher Secondary Physics (NCTB Class 11)Sample 1 • Orthogonal Vector Condition
Given vectors A = 2i + 3j - k and B = 3i - 2j + m k, determine the value of m for which the vectors are perpendicular.
Given Parameters:
- Vector A = 2î + 3ĵ - k̂
- Vector B = 3î - 2ĵ + m k̂
- Condition: A ⊥ B ⇒ A · B = 0
Procedural Solution:
Dot product formula: A · B = Ax·Bx + Ay·By + Az·Bz
Substituting: (2)(3) + (3)(-2) + (-1)(m) = 0
6 - 6 - m = 0
-m = 0 ⇒ m = 0
Final Answer:Required value of m = 0
Engineering Admission (BUET / JEE Main)Sample 2 • Cross Product & Parallelogram Area
Find the area of the parallelogram formed by adjacent side vectors A = i + 2j + 3k and B = 3i - 2j + k.
Given Parameters:
- Adjacent side A = î + 2ĵ + 3k̂
- Adjacent side B = 3î - 2ĵ + k̂
Procedural Solution:
Cross product A × B = |î ĵ k̂; 1 2 3; 3 -2 1|
= î(2×1 - 3×(-2)) - ĵ(1×1 - 3×3) + k̂(1×(-2) - 2×3)
= 8î + 8ĵ - 8k̂
Magnitude |A × B| = √(8² + 8² + (-8)²) = √(64 + 64 + 64) = √192 = 8√3 ≈ 13.856
Final Answer:Parallelogram area = 8√3 sq units (13.856 sq units)
HSC Board ExaminationSample 3 • Angle Between 3D Vectors
Determine the angle between vectors A = 2i + 2j - k and B = 6i - 3j + 2k.
Given Parameters:
- A = 2î + 2ĵ - k̂
- B = 6î - 3ĵ + 2k̂
Procedural Solution:
Dot product A · B = (2)(6) + (2)(-3) + (-1)(2) = 12 - 6 - 2 = 4
|A| = √(2² + 2² + (-1)²) = 3
|B| = √(6² + (-3)² + 2²) = 7
cos θ = (A · B) / (|A| |B|) = 4 / (3 × 7) = 4 / 21 ≈ 0.1905
θ = cos⁻¹(0.1905) ≈ 78.96°
Final Answer:Angle between vectors θ ≈ 78.96°
Board Exam & Laboratory Checklist
✓ Unit Conversion (SI)Convert all inputs to SI units prior to calculation.
✓ Formula & TranspositionExplicitly isolate target variable before substitution.
✓ Precision & RoundingPreserve required precision without premature truncation.
Practical Applications
How this scientific principle drives chemical engineering, aerospace, medicine, and research.
Frequently Asked Questions
Answers to common conceptual misconceptions and board examination guidelines.
When are two non-zero vectors orthogonal (perpendicular)?▼
Two non-zero vectors A and B are orthogonal if and only if their scalar dot product equals zero: A · B = |A||B| cos(90°) = 0. In algebraic components: AxBx + AyBy + AzBz = 0.
What is the condition for two vectors to be parallel?▼
Two vectors are parallel if their cross product equals the zero vector: A × B = 0, or if their corresponding Cartesian components are directly proportional: Ax/Bx = Ay/By = Az/Bz.
How is vector cross product used to find geometric areas?▼
The magnitude of the cross product |A × B| equals the area of the parallelogram spanned by vectors A and B. The area of the triangle formed by these two vectors is ½ |A × B|.