Vectors, 3D & Statistics
Vectors, dot and cross products, 3D geometry, probability, and statistics
Key Concepts
- Vector algebra: addition, scaling and components
- Dot product gives projection; cross product gives a perpendicular vector
- Scalar triple product gives the volume of a parallelepiped
- 3D geometry: equations of lines and planes
- Probability and statistics: Bayes' theorem, mean and variance
Important Formulae
| Dot product | a·b = |a||b| cosθ |
| Cross product | |a×b| = |a||b| sinθ (area of parallelogram) |
| Scalar triple product | [a b c] = a·(b×c) = volume |
| Conditional probability | P(A|B) = P(A∩B)/P(B) |
| Mean & variance | x̄ = Σx/n; σ² = Σ(x−x̄)²/n |
Quick Tips
- a·b = 0 means the vectors are perpendicular; a×b = 0 means they are parallel.
- If the scalar triple product is 0, the three vectors are coplanar.
- For a binomial distribution: mean = np, variance = npq.
Sample Practice Questions
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The dot product î · î is:
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Answer: 1
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The magnitude of the cross product |a × b| equals the area of the:
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Answer: Parallelogram formed by a and b
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The cross product î × ĵ is:
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Answer: k̂
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The dot product of two perpendicular vectors is:
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Answer: 0
Practice Questions
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What are the key concepts in Vectors, 3D & Statistics for JEE?
Vector algebra: addition, scaling and components; Dot product gives projection; cross product gives a perpendicular vector; Scalar triple product gives the volume of a parallelepiped; 3D geometry: equations of lines and planes; Probability and statistics: Bayes' theorem, mean and variance.
Which formulae are important for Vectors, 3D & Statistics in JEE?
Dot product: a·b = |a||b| cosθ; Cross product: |a×b| = |a||b| sinθ (area of parallelogram); Scalar triple product: [a b c] = a·(b×c) = volume; Conditional probability: P(A|B) = P(A∩B)/P(B); Mean & variance: x̄ = Σx/n; σ² = Σ(x−x̄)²/n.
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