Type two 3D vectors and watch A x B update as you go. You get magnitude (parallelogram area), a unit direction, angle readouts, and a sketch you can spin with drag or auto-rotate.
(0, 0, 0)Mechanics homework often hands you a force along one line and a position along another. The cross product packages how much turning leverage sits in the pair. Graphics pipelines use the same object when they need a surface normal from two edge directions.
A concrete statics example keeps the story grounded. Suppose a wrench handle runs from the origin to (0.18, 0, 0) meters and you push with (0, 120, 0) newtons at the tip. The moment vector about the bolt is r x F. You care about magnitude for fastener rating and direction for knowing which way the body wants to twist.
Numbers first, pictures second.
If you already trust the Dot Product Calculator for measuring alignment, think of the cross product as the companion that measures how far two directions sit out of plane with each other. When the angle hits 90°, |A x B| peaks for fixed lengths. When vectors line up, the parallelogram collapses to a line segment, so the area hits zero.
Simulation teams pair both tools with the Vector Calculator when they normalize inputs before feeding rigid-body solvers. The workflow sounds heavy, yet the user-facing task stays small: confirm components, multiply, read area and normal.
You supply A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃). The tool forms
A x B = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁).
Each row subtracts two products so the result is orthogonal to both inputs whenever the result is nonzero. Swap A and B and every component flips sign. Students lose points on exams for forgetting that single minus sign, so the preset buttons above include î x ĵ and parallel checks on purpose.
Determinant mnemonics help if you prefer geometry on paper. Expand the 3x3 matrix with unit vectors along the top row, components of A in the middle, components of B on the bottom, then expand minors carefully. The calculator skips the bookkeeping and still lists intermediate terms inside the collapsible algebra panel so you can match your scratch work line by line.
| Operation | Output type | Size idea |
|---|---|---|
| A · B | Scalar | |A||B| cos θ (projection story) |
| A x B | Vector | |A||B| sin θ (area story) |
Together they satisfy |A|²|B|² = (A · B)² + |A x B|² for 3D vectors, which is a useful sanity check when you are debugging handwritten notes.
Physics books switch between bold vectors, arrow hats, and column matrices without warning. The numeric work stays identical as long as you keep a consistent basis. Here every value is plain Cartesian, so you can paste homework components directly.
Engineers sometimes write ω x r for velocity in rotating frames, while mathematicians write the same structure when they define curl. The operation is stubbornly the same: antisymmetric product, length tied to sine, direction from a right-hand convention.
Try the (2,3,1)x(1,2,3) preset after you compute by hand once. If your middle component disagrees, recheck the y-slot formula: a₃b₁ - a₁b₃, not the cyclic copy of the x-slot. That single mix-up shows up on every third homework set.
We recommend treating a zero cross product as no unique normal direction, not as a broken calculation. Parallel vectors, duplicate directions, or a zero-length input all land here.
Floating-point noise might show tiny nonzero crumbs after you expect exact parallelism. For production physics code you would compare |A x B| against a tolerance tied to your units. This page shows raw arithmetic, so you still judge whether 10⁻¹² is noise or signal.
When homework jumps from two vectors to full 3x3 systems, the Matrix Inverse Calculator becomes the next stop. Cross products stay local; matrix tools handle simultaneous equations where several unknowns couple together.
The canvas is a teaching aid, not a CAD viewport. Perspective is simplified, labels omit depth cues on purpose, and extreme magnitudes only rescale visually inside the plot window.
Method: textbook 3D cross product on Cartesian components. Last reviewed: 20 March, 2026.
