Vector Cross Product Calculator

Calculator and formulas for computing the cross product of two 3D vectors

Vector Cross Product Calculator

3D Vector Cross Product

Calculates the cross product v₁ × v₂ of two 3D vectors, producing an orthogonal vector: v₁ × v₂ ⊥ v₁ and v₁ × v₂ ⊥ v₂

3D Vectors Only

The cross product is defined only for three-dimensional vectors. It produces a new vector perpendicular to both input vectors.

First Vector (v₁)
Second Vector (v₂)
Cross Product Result
X:
Y:
Z:
Cross Product Vector:
Magnitude |v₁ × v₂|:
The cross product produces a vector orthogonal to both input vectors.

Cross Product Info

Cross Product Properties

Orthogonal: Result is perpendicular to both vectors

v₁ × v₂ ⊥ v₁ v₁ × v₂ ⊥ v₂ 3D Only

Non-commutative: v₁ × v₂ = -(v₂ × v₁)
3D specific: Only defined for three-dimensional vectors

Right-Hand Rule
Thumb: Direction of v₁
Index Finger: Direction of v₂
Middle Finger: Direction of v₁ × v₂

Formulas for the Cross Product

Cross Product Formula
\[\vec{a} \times \vec{b} = \left[\matrix{a_2b_3 - a_3b_2\\a_3b_1 - a_1b_3\\a_1b_2 - a_2b_1}\right]\]

Component-wise calculation

Determinant Form
\[\vec{a} \times \vec{b} = \begin{vmatrix}\vec{i} & \vec{j} & \vec{k}\\a_1 & a_2 & a_3\\b_1 & b_2 & b_3\end{vmatrix}\]

Using unit vectors

Magnitude of Cross Product
\[|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta\]

Geometric interpretation

Parallelogram Area
\[A = |\vec{a} \times \vec{b}|\]

Area of the spanned parallelogram

Calculation Examples for Cross Product

Example 1: Standard Vectors
v₁ = [1, 2, 3], v₂ = [7, 8, 9]
\[\begin{aligned} v₁ \times v₂ &= \left[\matrix{2 \cdot 9 - 3 \cdot 8\\3 \cdot 7 - 1 \cdot 9\\1 \cdot 8 - 2 \cdot 7}\right] \\ &= \left[\matrix{18 - 24\\21 - 9\\8 - 14}\right] = \left[\matrix{-6\\12\\-6}\right] \end{aligned}\]

Result: [-6, 12, -6]

Example 2: With Magnitude
v₁ = [1, -4, 5], v₂ = [3, 5, 2]
\[\begin{aligned} v₁ \times v₂ &= \left[\matrix{(-4) \cdot 2 - 5 \cdot 5\\5 \cdot 3 - 1 \cdot 2\\1 \cdot 5 - (-4) \cdot 3}\right] \\ &= \left[\matrix{-8 - 25\\15 - 2\\5 + 12}\right] = \left[\matrix{-33\\13\\17}\right] \end{aligned}\]

|v₁ × v₂| = √((-33)² + 13² + 17²) = √1547 ≈ 39.33

Step-by-Step Calculation
X: a₂b₃ - a₃b₂
Y: a₃b₁ - a₁b₃
Z: a₁b₂ - a₂b₁

Each component is calculated according to the cross product formula

Important Properties
Anti-commutative

• v₁ × v₂ = -(v₂ × v₁)

• Order changes the sign

Orthogonality

• (v₁ × v₂) ⊥ v₁

• (v₁ × v₂) ⊥ v₂

Parallelism

• v × v = 0 (zero vector)

• Parallel vectors: v₁ × v₂ = 0

Applications of Cross Product

The cross product is fundamental in many areas of physics and engineering:

Mechanics & Physics
  • Torque: τ = r × F
  • Angular momentum: L = r × p
  • Lorentz force: F = q(v × B)
  • Surface normals and area calculations
Computer Graphics
  • Surface normals for lighting
  • Backface culling and visibility
  • 3D transformations and orientation
  • Collision detection and reflections
Geometry & CAD
  • Polygon area calculations
  • Surface and plane normals
  • Volume calculations (scalar triple product)
  • Orthogonal projections
Robotics & Navigation
  • Object orientation and rotation
  • Coordinate system transformations
  • Path planning and obstacle avoidance
  • Gyroscope and accelerometer sensors

Cross Product: Orthogonality in Three-Dimensional Space

The cross product is a unique operation of three-dimensional space that produces from two vectors a third vector perpendicular to both original vectors. This fundamental property makes it indispensable for applications ranging from classical mechanics through computer graphics to modern robotics. The geometric interpretation as the area of the spanned parallelogram connects algebraic calculation with intuitive geometry.

Summary

The cross product unites mathematical elegance with practical significance. The formula - systematic combination of vector components - guarantees orthogonal results and enables precise calculations of torques, surface normals, and orientations. From physics engines through CAD systems to robot control, the cross product remains an indispensable tool. It demonstrates how specialized mathematical operations elegantly solve complex spatial 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