How this instrument works
The cross product takes two vectors in three-dimensional space and returns a third: A × B = (ay·bz − az·by, az·bx − ax·bz, ax·by − ay·bx). Unlike the dot product, which collapses two vectors into a single number, this operation hands back a whole arrow — one pointing perpendicular to both A and B at once, tilted at a right angle to the plane the pair spans. Its direction follows the right-hand rule: lay a right hand flat along A, fold the fingers shut toward B, and the thumb springs up perpendicular to both, aimed along A × B.
The length of that result carries its own meaning. |A × B| equals |A||B|sinθ, which is exactly the area of the parallelogram with A and B as adjacent sides — a fact provable by dropping a perpendicular from the parallelogram's far corner and recognizing the classic base-times-height area formula in disguise. Swap the order of the operands and the sign flips, A × B = −(B × A), because reversing which vector the fingers curl from also reverses which way the thumb points; the cross product is anti-commutative rather than commutative like ordinary multiplication.
Two vectors pointing the same way, or exactly opposite ways, have no plane between them to be perpendicular to, so their cross product is the zero vector — sinθ is zero at both 0° and 180°. That degeneracy is a genuine edge case worth watching for: a computed torque, surface normal, or angular momentum that comes back as (0, 0, 0) is usually telling you the two inputs were parallel, not that something went wrong upstream.
- Enter the first arrow's three components into Vector A: x, Vector A: y, and Vector A: z.
- Enter the second arrow's three components into Vector B: x, Vector B: y, and Vector B: z.
- Read Result: x, Result: y, and Result: z together — they are the three components of the single vector A × B, not three separate answers.
- Check the sign of each result against the right-hand rule if the direction matters for your problem, such as a torque or a surface normal.
- Swap every value in Vector A with the matching value in Vector B to watch all three results flip sign, confirming the anti-commutative property.
Worked example — the two axis vectors î and ĵ
Set Vector A to (1, 0, 0), one unit along the x-axis, and Vector B to (0, 1, 0), one unit along the y-axis. Component by component: Result: x = ay·bz − az·by = 0·0 − 0·1 = 0; Result: y = az·bx − ax·bz = 0·0 − 1·0 = 0; Result: z = ax·by − ay·bx = 1·1 − 0·0 = 1. The sheet reports (0, 0, 1) — one unit straight up the z-axis, and nothing else.
That answer, î × ĵ = k̂, is the defining relationship behind every right-handed coordinate system taught in a first physics or linear-algebra course, so it doubles as a trustworthy check on any cross-product implementation. Note what did not happen: a dot product on these same two inputs would return the single number 0, since the vectors are perpendicular. The cross product instead returns an entire third vector, k̂, perpendicular to both originals — a genuinely different kind of answer from a genuinely different operation.
Questions
What does the cross product actually give you?
A third vector, perpendicular to both inputs, whose length equals the area of the parallelogram the two inputs span and whose direction follows the right-hand rule. It is not a single number — that is the dot product's job — and it exists only for vectors in three-dimensional space, unlike the dot product, which works in any number of dimensions.
How do I figure out which way the result vector points?
Lay your right hand flat with fingers stretched along Vector A, then fold them shut toward Vector B, sweeping through the smaller of the two angles between them. Your thumb springs up perpendicular to both, aimed along A × B. Reverse the sweep — folding from B toward A instead — and the thumb flips to the opposite side, which is exactly why A × B and B × A share a length but point opposite ways.
Why does the calculator return (0, 0, 0) sometimes?
A zero result means the two input vectors are parallel or anti-parallel — pointing along the same line, in the same direction or the opposite one. There is no unique plane spanned by two parallel arrows, so sinθ is zero and every component of the cross product collapses to zero along with it.
How is the cross product different from the dot product?
The dot product ax·bx + ay·by + az·bz returns a single scaled-by-cosθ number describing how aligned two vectors are; the cross product returns a whole vector, scaled by sinθ, describing the plane they span and the direction perpendicular to it. One shrinks two arrows to a scalar; the other grows them into a third arrow.
Does swapping Vector A and Vector B change the answer?
Yes — every component flips sign, since A × B = −(B × A). The length and the line the result lies on stay identical; only the direction along that line reverses, a direct consequence of the right-hand rule pointing the opposite way once the curl order is reversed.
Where does this calculation actually get used?
Torque equals a lever-arm vector crossed with a force vector; angular momentum equals a position vector crossed with a momentum vector; and computer graphics engines cross two edge vectors of a triangle to find the surface normal used for lighting. In every case, the useful output is specifically a direction perpendicular to two known ones.