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DIY Cost Tools

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Cross Product Calculator

Find a vector perpendicular to two three-dimensional vectors, or check the orientation of a coordinate calculation. The order of the operands matters. Enter A first and B second for A × B. Use the same right-handed orthonormal coordinate system for both. The output components describe the new vector; its magnitude can also represent the area of the parallelogram spanned by the inputs.

Enter every field, including explicit zeros. Decimal and scientific notation are accepted; commas, fractions and blank fields are rejected. Each input must be finite and no larger than 1 billion in absolute value.

Result x0
Result y0
Result z1
Result magnitude1

How to use this calculator

  1. Match each input to its labeled coordinate, coefficient or matrix position.
  2. Check the signs and use one consistent coordinate convention.
  3. Select Calculate, or press Enter, and read the labeled results.

Formula and method

A × B = (AyBz − AzBy, AzBx − AxBz, AxBy − AyBx).

The calculation runs locally in your browser. Changing an input requires selecting Calculate again. If an input or mathematical condition is invalid, the previous result is hidden instead of leaving a stale answer visible.

Worked example

For A = (1, 0, 0) and B = (0, 1, 0), A × B = (0, 0, 1), with magnitude 1.

Limitations and interpretation

Swapping A and B reverses the result. Parallel inputs, including a zero vector, give a zero cross product. This tool is for ordinary three-dimensional real vectors. It does not validate a physical force diagram or establish a safe mechanical design.

Results are displayed to approximately 10 significant digits. The underlying calculation uses ordinary floating-point numbers and is not exact symbolic arithmetic. Rounded results can differ slightly from hand calculations. For a measured quantity, choose precision that reflects the measurements rather than all the digits shown. Verify consequential calculations independently.

Frequently asked questions

Why does reversing the vectors change the answer?

The cross product is ordered: B × A = −(A × B).

Can I enter negative or decimal values?

Yes, provided the values meet the mathematical conditions described above. Type a normal hyphen for a negative sign. Do not type units, commas or a fraction such as 1/2; enter 0.5 instead.

Why did the calculator hide my previous result?

A missing or invalid input, undefined operation or unstable numerical case produces an explanation. Correct the input and calculate again to obtain a result for the current values.

Review notes and reference

Reviewed October 3, 2026. The stated default example and a boundary or algebraic check were tested against expected values. Input validation checks included blank fields, nonnumeric text and values outside the supported range. Wolfram MathWorld formula reference provides mathematical background. This tool is for calculation and education; its output alone does not validate a professional design.

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