Cross Product Calculator

Compute the 3D cross product a×b and its magnitude.

VectorsLinear AlgebraEngineering Math

Calculator

a₁
a₂
a₃
b₁
b₂
b₃
Please enter valid values.

What this calculator teaches

The cross product produces a vector perpendicular to two 3D vectors.

Its magnitude measures the area of the parallelogram spanned by the vectors.

Formula & Symbols

ConceptFormulaMeaning
Cross producta×b=(a₂b₃-a₃b₂, a₃b₁-a₁b₃, a₁b₂-a₂b₁)Vector perpendicular to a and b.
Magnitude|a×b|=|a||b|sinθArea interpretation.

Step-by-Step Examples

Example 1
i × j
  • a=(1,0,0), b=(0,1,0).
  • Apply the component formula.
✓ a×b=(0,0,1).

Where students use this

🎓
College mathematics
Use the tool to connect notation, formulas, and numeric answers.
🧪
Engineering analysis
Apply the result to models, systems, approximations, and design calculations.
💻
Computer science
Use matrix, vector, numerical, and optimization ideas in algorithms and data work.

Common Mistakes to Avoid

⚠️
Ignoring restrictions

Check denominator, domain, endpoint, and matrix-invertibility conditions before trusting a result.

⚠️
Rounding too early

Carry extra decimals through intermediate steps and round only the final answer.

⚠️
Mixing symbolic and numeric answers

Some topics need exact symbolic reasoning, while this calculator gives a clear numerical learning check.

Frequently Asked Questions

Is this calculator meant for students? ▾
Yes. It gives an answer and shows the main method so students can learn the reasoning instead of copying only the result.
Can I use decimals and negative numbers? ▾
Yes, when the topic allows them. Some formulas have restrictions such as positive radii, nonzero denominators, or valid intervals.
Why can textbook answers look different? ▾
Equivalent algebraic forms, different rounding, or using exact notation instead of decimals can make answers look different while meaning the same thing.

Related Math Calculators

🔢 Dot Product Calculator🔢 Vector Projection Calculator🔢 Equation of a Plane Calculator

Formula Explorer connections

Interpretation: This relationship represents vectors, linear systems or transformations using matrix structure and vector operations. Assumption: Matrix dimensions must be compatible. Inversion, decomposition and unique solutions require rank, conditioning or nonzero-determinant conditions.

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