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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

  1. The dot product of two perpendicular vectors is:

    • 0
    • 1
    • Their magnitudes multiplied
    • Undefined
    Show answer

    Answer: 0

  2. The magnitude of the cross product |a × b| equals the area of the:

    • Triangle formed by a and b
    • Parallelogram formed by a and b
    • Circle of radius |a|
    • Square of side |a|
    Show answer

    Answer: Parallelogram formed by a and b

  3. The cross product î × ĵ is:

    • k̂
    • −k̂
    • 0
    • î
    Show answer

    Answer: k̂

  4. For three coplanar vectors, the scalar triple product is:

    • 0
    • 1
    • Their sum
    • Undefined
    Show answer

    Answer: 0

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Practice Questions

Practise randomly selected JEE questions on Vectors, 3D & Statistics. Answers are revealed after each question.

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