Vector Multiplication

Calculator and formulas for component-wise multiplication of 2D, 3D and 4D vectors

Vector Multiplication Calculator

Component-wise Vector Multiplication

Multiplies two vectors v₁ × v₂ through component-wise multiplication: [x₁×x₂, y₁×y₂, z₁×z₂]

Not to be confused with Dot or Cross Product

This is an element-wise multiplication (Hadamard product). For dot product or cross product, use the corresponding other calculators.

Select Vector Dimension
First Factor (v₁)
Second Factor (v₂)
Multiplication Result
X:
Y:
Z:
W:
Product Vector:
Components are multiplied pairwise: v₁ × v₂ = [x₁×x₂, y₁×y₂, ...]

Multiplication Info

Element-wise Properties

Hadamard Product: Corresponding components are multiplied

Commutative Associative Element-wise

Difference: Not dot or cross product
Result: New vector of same dimension

Examples
[2, 3, 4] × [5, 4, 3] = [10, 12, 12]
[1, 2] × [3, 4] = [3, 8]
[x, y, z] × [1, 1, 1] = [x, y, z]

Formulas for Vector Multiplication

2D Element-wise Multiplication
\[\left[\matrix{a\\b}\right] \circ \left[\matrix{c\\d}\right] = \left[\matrix{a \cdot c\\b \cdot d}\right]\]

Hadamard product in the plane

3D Element-wise Multiplication
\[\left[\matrix{a\\b\\c}\right] \circ \left[\matrix{x\\y\\z}\right] = \left[\matrix{a \cdot x\\b \cdot y\\c \cdot z}\right]\]

Components multiplied pairwise

4D Element-wise Multiplication
\[\left[\matrix{a\\b\\c\\d}\right] \circ \left[\matrix{w\\x\\y\\z}\right] = \left[\matrix{a \cdot w\\b \cdot x\\c \cdot y\\d \cdot z}\right]\]

Higher-dimensional element-wise multiplication

General Rule
\[(\vec{v_1} \circ \vec{v_2})_i = (v_1)_i \cdot (v_2)_i\]

Component-wise multiplication

Calculation Examples for Vector Multiplication

Example 1: 3D Multiplication
[2, 3, 4] × [5, 4, 3]
\[\vec{v_1} \circ \vec{v_2} = \left[\matrix{2 \cdot 5\\3 \cdot 4\\4 \cdot 3}\right] = \left[\matrix{10\\12\\12}\right]\]

Result: [10, 12, 12]

Example 2: 2D Multiplication
[6, -4] × [2, 3]
\[\vec{u} \circ \vec{w} = \left[\matrix{6 \cdot 2\\-4 \cdot 3}\right] = \left[\matrix{12\\-12}\right]\]

Result: [12, -12]

Step-by-Step Calculation
X: 2 × 5 = 10
Y: 3 × 4 = 12
Z: 4 × 3 = 12

Each component is multiplied separately

Comparison of Different Vector Products
Element-wise (×)

• [a, b] × [c, d] = [ac, bd]

• Same dimension

• Hadamard product

Dot Product (·)

• [a, b] · [c, d] = ac + bd

• Result: Scalar

• Dot-product

Cross Product (×)

• Only for 3D vectors

• Result: Orthogonal vector

• Cross-product

Applications of Element-wise Multiplication

Element-wise multiplication finds application in various technical and scientific fields:

Data Processing & Signals
  • Signal filtering and weighting
  • Window functions in spectral analysis
  • Masking operations in images
  • Point-wise scaling of datasets
Machine Learning
  • Neural networks: activation functions
  • Feature-wise weightings
  • Element-wise attention mechanisms
  • Dropout and regularization
Computer Graphics
  • RGB color mixing and modulation
  • Texture blending and masking
  • Per-component lighting calculations
  • Alpha blending and transparency
Science & Simulation
  • Component-wise scaling of physical quantities
  • Finite-element methods
  • Statistical weightings
  • Coordinate transformations

Element-wise Multiplication: The Hadamard Product

Element-wise vector multiplication, also called the Hadamard product, is a fundamental operation that multiplies corresponding components of two vectors with each other. Unlike dot or cross products, it produces a new vector of the same dimension. This operation is particularly valuable in digital signal processing, machine learning, and computer graphics, where point-wise operations are frequently needed.

Summary

Element-wise multiplication extends vector operations with a practical, intuitive method for component-wise combination. The simple rule - multiply corresponding components - enables efficient filtering, weighting, and modulation in various application domains. From image processing through neural networks to scientific simulation, the Hadamard product offers a direct method for point-wise manipulation of vector data. It shows how simple mathematical operations can elegantly solve complex practical problems.




Vector Functions

Addition • Subtraction • Multiplication • Scalar Multiplication • Division • Scalar Division • Dot Product • Cross Product • Interpolation • Distance • Distance Squaret • Normalization • Reflection • Magnitude • Squared-Magnitude • Triple-Product