डॉट गुणनफल कैलकुलेटर — सूत्र

Dot Product

Formula

A · B = a₁b₁ + a₂b₂ + a₃b₃

Geometric Interpretation

A · B = A×B× cos(theta)

So: theta = arccos(A · B / (A×B))

Properties

  • If A · B = 0, the vectors are perpendicular
  • If A · B > 0, the angle is acute (less than 90°)
  • If A · B < 0, the angle is obtuse (greater than 90°)

    • हल किया गया उदाहरण

    Dot product of (1,2,3) and (4,5,6).

    1. A · B = 1×4 + 2×5 + 3×6 = 4 + 10 + 18 = 32
    2. |A| = √(1+4+9) = √14 ≈ 3.742
    3. |B| = √(16+25+36) = √77 ≈ 8.775
    4. cos(theta) = 32/(3.742 × 8.775) ≈ 0.9746
    5. theta ≈ 12.93°