Calcolatore Prodotto Scalare

Calcola il prodotto scalare (dot product) di due vettori. Vettori 2D e 3D.

Dot

32.0000

Mag A3.7417
Mag B8.7750
Angle12.9332 deg
Vedi formula →

Formula

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°)

    • Esempio Risolto

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

    1. 01A · B = 1×4 + 2×5 + 3×6 = 4 + 10 + 18 = 32
    2. 02|A| = √(1+4+9) = √14 ≈ 3.742
    3. 03|B| = √(16+25+36) = √77 ≈ 8.775
    4. 04cos(theta) = 32/(3.742 × 8.775) ≈ 0.9746
    5. 05theta ≈ 12.93°

    Domande Frequenti

    What is the dot product?

    The dot product is a way to multiply two vectors to get a scalar (single number). It measures how much two vectors point in the same direction.

    When is the dot product zero?

    The dot product is zero when the vectors are perpendicular (orthogonal). This is a common test for perpendicularity.

    What is the difference between dot product and cross product?

    The dot product gives a scalar and measures alignment. The cross product gives a vector perpendicular to both inputs and measures the area of the parallelogram they span.

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