Vector Division

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

Vector Division Calculator

Component-wise Vector Division

Divides two vectors v₁ ÷ v₂ through component-wise division: [x₁/x₂, y₁/y₂, z₁/z₂]

Select Vector Dimension
Dividend (v₁)
Divisor (v₂)
Division Result
X:
Y:
Z:
W:
Quotient Vector:
Components are divided individually: v₁ ÷ v₂ = [x₁/x₂, y₁/y₂, ...]

Vector Division Info

Division Properties

Component-wise: Corresponding components are divided

Non-commutative Watch for zeros Element-wise

Caution: Division by zero not allowed
Result: New vector of same dimension

Example
[5, 4, 9] ÷ [2, 2, 3] = [2.5, 2, 3]
[8, -6] ÷ [4, 2] = [2, -3]
[x, y, z] ÷ [1, 1, 1] = [x, y, z]

Formulas for Vector Division

2D Vector Division
\[\left[\matrix{a\\b}\right] \div \left[\matrix{c\\d}\right] = \left[\matrix{a/c\\b/d}\right]\]

Division in the plane

3D Vector Division
\[\left[\matrix{a\\b\\c}\right] \div \left[\matrix{x\\y\\z}\right] = \left[\matrix{a/x\\b/y\\c/z}\right]\]

Division in space

4D Vector Division
\[\left[\matrix{a\\b\\c\\d}\right] \div \left[\matrix{w\\x\\y\\z}\right] = \left[\matrix{a/w\\b/x\\c/y\\d/z}\right]\]

Higher-dimensional division

General Rule
\[\vec{v_1} \div \vec{v_2} = \left[\frac{v_{1i}}{v_{2i}}\right]_{i=1}^n\]

Component-wise division

Calculation Examples for Vector Division

Example 1: 3D Division
[5, 4, 9] ÷ [2, 2, 3]
\[\vec{v_1} \div \vec{v_2} = \left[\matrix{5/2\\4/2\\9/3}\right] = \left[\matrix{2.5\\2\\3}\right]\]

Result: [2.5, 2, 3]

Example 2: 2D Division
[8, -12] ÷ [4, 3]
\[\vec{u} \div \vec{w} = \left[\matrix{8/4\\-12/3}\right] = \left[\matrix{2\\-4}\right]\]

Result: [2, -4]

Step-by-Step Calculation
X: 5 ÷ 2 = 2.5
Y: 4 ÷ 2 = 2
Z: 9 ÷ 3 = 3

Each component is divided separately

Important Note: Division by Zero
Invalid

• [4, 6] ÷ [2, 0] → Error

• [x, y, z] ÷ [a, 0, b] → Error

• No division by zero vector

Valid

• [4, 6] ÷ [2, 3] = [2, 2]

• [x, y] ÷ [1, 1] = [x, y]

• All divisor components ≠ 0

Applications of Vector Division

Vector division is applied in various technical and scientific fields:

Data Analysis & Statistics
  • Normalization of data vectors
  • Ratio calculations between categories
  • Scaling and standardization
  • Index and metrics calculation
Computer Graphics
  • Texture coordinate scaling
  • Viewport transformations
  • UV mapping adjustments
  • Screen-space calculations
Engineering
  • Stress-strain ratios
  • Scaling factors in CAD
  • Proportional load distribution
  • Dimensional adjustments
Science
  • Relative concentrations
  • Ratios of physical quantities
  • Normalized measurement series
  • Proportionality factors

Vector Division: Component-wise Operation with Restrictions

Vector division is a component-wise operation that, unlike addition and multiplication, requires special precautions. Since division by zero is mathematically undefined, all components of the divisor vector must be non-zero. This operation finds primary application in normalization, scaling, and calculating ratios between corresponding vector components.

Summary

Vector division extends the fundamental operations of linear algebra with a practical, though restricted, operation. The simple rule - divide corresponding components - enables efficient scaling and normalization across various application domains. From data analysis through computer graphics to engineering, vector division provides a direct method for computing component-wise ratios. The need for zero-checking demonstrates how mathematical precision makes practical applications more robust.

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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