Cross Product Calculator | Compute Vector Cross Product

Formula: A × B = (Ay·Bz − Az·By, Az·Bx − Ax·Bz, Ax·By − Ay·Bx)

The Cross Product Calculator is built on a well-established calculation method. It uses the formula A × B = (Ay·Bz − Az·By, Az·Bx − Ax·Bz, Ax·By − Ay·Bx) to turn your inputs into a reliable result. Cross Product Calculator is a free online calculator that helps you work out cross product quickly and accurately, right in your browser. The steps are shown on the page so you can follow the reasoning from input to output.

The cross product of two 3D vectors returns a third vector perpendicular to both, with length equal to the area of the parallelogram they span.

1. Enter the x, y and z components of vector A.
2. Enter the x, y and z components of vector B.
3. The calculator applies the determinant formula for each component.
4. Read the resulting vector and its magnitude.

Worked example: (2, 3, 4) × (5, 6, 7) = (3·7−4·6, 4·5−2·7, 2·6−3·5) = (−3, 6, −3).

What is the Cross Product Calculator?

The Cross Product Calculator is a free, browser-based math calculator tool that helps you Cross Product Calculator is a free online calculator that helps you work out cross product quickly and accurately, right in your browser.. Instead of working through the math by hand or in a spreadsheet, you enter your values and the calculator returns an accurate result instantly — while still showing the formula and the steps so you can verify the reasoning. It is designed for quick everyday use: no sign-up, no installation, and everything runs locally in your browser for complete privacy.

How to use the Cross Product Calculator

1. Enter the required values into the input fields.
2. Press the calculate button — the result appears immediately, updated live as you change any value.
3. Read the step-by-step breakdown below the result to see exactly how the calculation was performed.
4. Use the interpretation guide to understand what the result means for your situation, and try different inputs to see how they change the outcome.

How to use the Cross Product Calculator

Enter the components of two 3D vectors A = (Ax, Ay, Az) and B = (Bx, By, Bz), and the tool computes the cross product with the formula A × B = (Ay·Bz − Az·By, Az·Bx − Ax·Bz, Ax·By − Ay·Bx). The result is a new vector perpendicular to both inputs. The tool also returns the magnitude of the cross product, which equals the area of the parallelogram the two vectors span.

Interpreting your result

The cross product is the workhorse of 3D geometry and physics: it produces the normal vector to a plane, the torque from a force at a distance, the angular momentum of a rotating body, and the magnetic force on a moving charge. Its direction follows the right-hand rule — curl your fingers from A toward B and your thumb points along A × B — so swapping the inputs flips the result's sign. The magnitude |A × B| = |A||B|sin(θ) also gives the parallelogram area, which makes it the geometric tool for finding areas and volumes in 3D space.

Common mistakes to avoid

The most common error is mixing up the component order in the formula — the pattern Ay·Bz − Az·By, then Az·Bx − Ax·Bz, then Ax·By − Ay·Bx — one wrong pairing changes every component. Second, forgetting that the cross product is anti-commutative: A × B = −(B × A), so the order of inputs matters. Third, confusing it with the dot product, which returns a scalar; the cross product only exists in 3D (and 7D) and returns a vector. Finally, using the right-hand rule backwards flips the direction of the normal vector.

Tips for best results

To find a plane's normal vector, cross two non-parallel vectors lying in the plane — the order determines which side the normal faces. For torque problems, cross the position vector with the force vector, in that order. Use the magnitude to compute parallelogram areas and, with a third vector, the volume of a parallelepiped via the scalar triple product. Check perpendicularity: the dot product of the result with either input must be zero. Verify your right-hand rule before trusting a direction in physics problems.

Authoritative source: Wolfram MathWorld