Dot Product Calculator

  • Enter vectors A and B, and select the operation.
  • Click "Calculate" to perform the calculation.
  • View the result, calculation details, and history below.
  • Click "Clear" to reset the inputs and results.
  • Click "Copy" to copy the result to the clipboard.


Calculation Details:

Calculation History:

    What is Dot Product?

    The dot product, also known as the scalar product or inner product, is a mathematical operation that takes two vectors and produces a scalar (a single numerical value). It is a fundamental operation in linear algebra and vector calculus. The dot product is denoted using a dot (·) or sometimes a cross (×) between two vectors.

    All Formulae Related to Dot Product

    Here are the key formulae related to the dot product:

    Formula for Dot Product (Scalar Product): The dot product of two vectors, A and B, can be calculated using the magnitudes of the vectors and the cosine of the angle (θ) between them:

    A · B = |A| * |B| * cos(θ)

    In this formula:

    • A · B represents the dot product of vectors A and B.
    • |A| is the magnitude (length) of vector A.
    • |B| is the magnitude (length) of vector B.
    • θ is the angle between vectors A and B.

    Dot Product Using Components: The dot product can also be calculated using the components of the vectors. For vectors in three-dimensional space (with components A₁, A₂, A₃ and B₁, B₂, B₃):

    A · B = A₁ * B₁ + A₂ * B₂ + A₃ * B₃

    Properties of the Dot Product: Several properties are associated with the dot product:

    1. Commutative: The dot product is commutative, meaning A · B = B · A.
    2. Distributive: The dot product is distributive over addition, meaning A · (B + C) = A · B + A · C.
    3. Scalar Multiplication: The dot product can be distributed across scalar multiplication, meaning k(A · B) = (kA) · B = A · (kB), where k is a scalar.
    4. Zero Dot Product: If the dot product of two vectors is zero, it indicates that the vectors are orthogonal (perpendicular) to each other.
    5. Parallel Vectors: If two vectors are parallel, the dot product is equal to the product of their magnitudes: A · B = |A| * |B|.
    6. Angle Calculation: The angle (θ) between two vectors can be calculated using the dot product and trigonometry:
      • cos(θ) = (A · B) / (|A| * |B|)

    Applications of Dot Product Calculator in Various Fields

    Here are some common applications of a dot product calculator in different domains:

    1. Physics and Engineering:
      • Mechanics: Calculating work done, torque, and moment of a force using the dot product of vectors.
      • Electromagnetism: Analyzing electric and magnetic fields, calculating electric flux, and magnetic flux.
    2. Computer Graphics and Animation:
      • Lighting and Shading: Determining the intensity of light at a point on a surface by calculating the dot product between the surface normal and the light direction.
      • 3D Transformations: Performing rotations, translations, and scaling operations in 3D graphics.
    3. Mechanical Engineering:
      • Structural Analysis: Evaluating stresses and strains in materials and structures using force vectors.
      • Machine Design: Analyzing forces and moments in mechanical systems.
    4. Electrical Engineering:
      • Circuit Analysis: Calculating power and energy in electrical circuits using voltage and current vectors.
      • Signal Processing: Applying filters and performing vector signal processing operations.
    5. Aerospace and Aeronautics:
      • Aircraft Dynamics: Analyzing forces and moments acting on aircraft and spacecraft.
      • Navigation and Flight Path Planning: Determining position and direction in three-dimensional space.
    6. Robotics and Automation:
      • Robot Kinematics: Solving forward and inverse kinematics problems for robot manipulators.
      • Control Systems: Implementing control algorithms for robot motion and manipulation.

    Benefits of Using the Dot Product Calculator

    Using a dot product calculator offers several benefits across various fields and applications, especially in situations where precise vector calculations are required. Here are the key advantages of using a dot product calculator:

    1. Accuracy: Dot product calculators provide highly accurate results, reducing the potential for human calculation errors associated with manual computations.
    2. Efficiency: They perform dot product calculations quickly and efficiently, saving time, especially when dealing with a large number of vectors or complex vector operations.
    3. Consistency: Dot product calculators ensure consistent and standardized calculations, eliminating discrepancies that can arise from manual calculations.
    4. Versatility: These calculators are versatile tools suitable for use in various fields, including physics, engineering, computer graphics, and mathematics.
    5. Complex Vector Operations: Dot product calculators simplify complex vector operations, allowing users to focus on solving problems and making decisions rather than performing tedious calculations.


    1. “Beyond Projections: Dot Products in Linear Algebra and Machine Learning” by SIAM Journal on Matrix Analysis and Applications
    2. “From Physics to Computer Graphics: Dot Products in Force Calculations and Lighting Models” by Journal of Graphics, Electronics, and Vision

    Last Updated : 27 February, 2024

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    21 thoughts on “Dot Product Calculator”

    1. This article provides a comprehensive understanding of the dot product and its mathematical properties

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