The cross product calculator above returns a × b for two vectors in three dimensions, together with the quantities that make the cross product worth computing: its magnitude, the area of the parallelogram the two vectors span, the triangle area that is half of it, the unit vector perpendicular to both, the angle between the inputs, and the scalar triple product with an optional third vector. Every one of those is a different reading of the same operation, and each has its own use.
Arb Digital publishes free calculators for the arithmetic that is easy to do and easy to get subtly wrong. The cross product is the clearest example in vector algebra: one sign error in the middle component produces a vector of plausible length pointing somewhere completely different, and nothing about the answer looks suspicious. Showing the magnitude, the angle, the unit normal and the areas at the same time makes such an error visible, because those quantities have to agree with each other.
What This Cross Product Calculator Does, and How It Differs From Our Other Vector Tools
Three of our live tools touch this territory, and it is worth being explicit about the boundaries so you land on the right one. Our vector calculator is a general workbench: it does addition, subtraction, scalar multiples, dot, cross and projection in two or three dimensions, one operation at a time. It computes a cross product but stops at the vector and its magnitude. Our cosine similarity calculator does not compute a cross product at all — it uses the dot product to measure how closely two vectors point the same way, in any number of dimensions, and it deliberately discards magnitude. Our companion dot product calculator handles the projection-and-work side of vector multiplication.
This page is the cross product taken further than a general tool goes. It reports the parallelogram and triangle areas the magnitude measures, the unit normal that defines an orientation, the scalar triple product that tests coplanarity and gives a signed volume, and a torque mode that labels the inputs as position and force. If you want one cross product among several other vector operations, the vector calculator is quicker. If the cross product is the point — because you need an area, a normal direction, a torque or a volume — this is the page built for it.
How to Use It
- Type each vector as three numbers. Commas or spaces both work. The cross product is defined in three dimensions only, so all three components are required.
- Get the order right. a × b and b × a have the same length and opposite directions. Use the swap button to see the sign flip rather than retyping.
- Choose a mode if it helps. Torque mode relabels the inputs as a position vector and a force, which is the arrangement most people actually need this for.
- Add a third vector for volume work. The triple product cell gives the signed volume of the parallelepiped spanned by all three, and it is zero exactly when they are coplanar.
- Sanity-check with the angle. The magnitude equals |a||b|sinθ, so an angle near zero or 180 degrees must produce a near-zero cross product.
The Formula and How It's Calculated
The cross product of a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃) is (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). The tidiest way to remember it is as the symbolic determinant of a 3 × 3 matrix whose first row is the unit vectors i, j, k, whose second row is a and whose third row is b. Expanding along the top row gives the three components directly, with the minus sign on the middle term supplied automatically by the cofactor pattern — which is where most hand calculations go wrong. Our determinant calculator evaluates the numeric determinants that expansion produces.
Take the defaults, a = (3, −3, 1) and b = (4, 9, 2). The first component is (−3)(2) − (1)(9) = −15. The second is (1)(4) − (3)(2) = −2. The third is (3)(9) − (−3)(4) = 39. So a × b = (−15, −2, 39), and its magnitude is √(225 + 4 + 1521) = √1750 = 41.8330. Check it against the trigonometric form: |a| = √19 = 4.3589, |b| = √101 = 10.0499, and the angle between them works out at 107.26 degrees, whose sine is 0.9550. Multiply and you get 4.3589 × 10.0499 × 0.9550 = 41.833, the same number by a completely different route. Lamar University's Calculus II notes on the cross product set out the determinant form and its basic properties.
The Right-Hand Rule
The formula fixes the length of a × b and fixes which of the two perpendicular directions it takes. That choice is a convention, and the right-hand rule is how you read it off. Point the fingers of your right hand along a, curl them toward b through the smaller angle, and your extended thumb points along a × b. Do the same with your left hand and you get the opposite answer, which is why the handedness matters and why the rule is always stated for the right hand.
The diagram shows the three quantities together: a and b spanning a parallelogram, the angle between them, and the result standing perpendicular to that whole plane with a length equal to the parallelogram's area. This is why the cross product is called anticommutative rather than merely non-commutative — swapping the inputs does not give an unrelated answer, it gives exactly the negative, because you are now curling your fingers the other way. Press the swap button above and watch all three components change sign while the magnitude, the areas and the angle stay put.
Area of a Parallelogram and a Triangle
The magnitude |a × b| = |a||b|sinθ is precisely the area of the parallelogram with a and b as adjacent edges, because |b|sinθ is the perpendicular height when a is treated as the base. Half of that is the area of the triangle with those two edges, which makes the cross product the most reliable way to get a triangle's area from coordinates — no need to find a base, drop a perpendicular, or worry about whether the triangle is obtuse.
For a triangle with vertices P, Q and R, form the edge vectors Q − P and R − P and take half the magnitude of their cross product. The method works identically in three dimensions, where the triangle is tilted in space and elementary formulas become awkward. In two dimensions only the z-component survives, and half its absolute value is the familiar shoelace formula. Our triangle area calculator covers the side-length and base-height routes for the plane case.
Torque: Why It Is r × F and Never F × r
Torque is the classic physical cross product: τ = r × F, where r is the position vector from the pivot to the point where the force is applied, and F is the force. The magnitude |r||F|sinθ captures the everyday fact that a force pushing straight along the lever arm does nothing, while a force perpendicular to it does the most possible. The direction, given by the right-hand rule, encodes the axis and the sense of rotation.
The order is not arbitrary and the mnemonic is worth fixing: the position vector comes first. Writing F × r reverses the sign, which in a rotational dynamics problem means the body spins the wrong way. A worked case: a spanner 0.3 m long lying along the x-axis, with a 150 N force applied straight up the y-axis, gives r = (0.3, 0, 0) and F = (0, 150, 0). The cross product is (0, 0, 45), so the torque is 45 newton-metres about the z-axis, counter-clockwise when viewed from positive z. Tilt the force 30 degrees off perpendicular and the magnitude drops to 45 sin 60° = 38.97 N·m. Our torque calculator handles the scalar force-times-distance form when the geometry is simple enough not to need vectors.
The Triple Product, Coplanarity and Volume
The scalar triple product (a × b) · c gives the signed volume of the parallelepiped with a, b and c as edges. The reasoning is direct: |a × b| is the area of the base, and dotting with c projects c onto the normal direction to give the perpendicular height. Volume is base times height, and the sign records whether the three vectors form a right-handed or a left-handed set.
The most useful consequence is the coplanarity test. If the triple product is zero, c lies in the plane of a and b, so all three vectors are coplanar and the parallelepiped is flat. That is a far cleaner test than trying to solve for a plane equation and checking membership. With the defaults, a × b = (−15, −2, 39) and c = (1, 2, 3), giving −15 − 4 + 117 = 98, so those three vectors are not coplanar and span a volume of 98 cubic units. Wolfram MathWorld's entry on the cross product lists the identities that follow, including the cyclic symmetry that lets you rotate a, b and c without changing the value.
Why the Cross Product Only Exists in Three Dimensions
In two dimensions there is no direction perpendicular to the plane to point in, so the cross product of two plane vectors is usually treated as the scalar a₁b₂ − a₂b₁ — the z-component of the three-dimensional result with the other two components zero. That scalar still gives the signed parallelogram area, which is why the shoelace formula works, but it is not a vector.
In four dimensions and above the construction fails outright, because the space perpendicular to two given vectors is more than one dimension wide and there is no canonical choice within it. A vector-valued product with the cross product's properties exists only in three and seven dimensions, which is a genuinely surprising algebraic fact rather than a limitation of notation. The general replacement is the wedge product, which returns an oriented plane element rather than a vector and works in any dimension. For similarity work in high-dimensional spaces the operation that survives is the dot product, which is what our cosine similarity calculator uses on embedding vectors with hundreds of components.
Arb Digital's web team builds interactive tools, configurators and visualisations where the maths has to hold up under real use.
See Our Web Design Work Talk To Our TeamCommon Mistakes to Avoid
- Dropping the sign on the middle component — the j term is a₃b₁ − a₁b₃, which is the reverse of the pattern used for i and k. This single slip accounts for most wrong answers.
- Reversing the order — b × a is the exact negative of a × b, so in a torque or angular-momentum problem it reverses the direction of rotation.
- Expecting a scalar — the cross product returns a vector. The scalar that measures alignment is the dot product, and confusing the two is a category error rather than an arithmetic one.
- Reading a zero result as an error — zero means the vectors are parallel, which is genuine information: a force acting along the lever arm produces no torque at all.
- Using it in two dimensions as though it returns a vector — with plane vectors the result has only a z-component, and treating that scalar as a vector in the plane is meaningless.
Related Free Tools From Arb Digital
Run several vector operations side by side with the vector calculator, handle projections and work done with the dot product calculator, compare direction in high dimensions with the cosine similarity calculator, get scalar torque from force and distance with the torque calculator, or find a plane triangle's area from its sides with the triangle area calculator. The full free online tools hub lists every mathematics tool we publish.
Frequently Asked Questions
It is a third vector perpendicular to both inputs, whose length equals the area of the parallelogram they span. Its direction is fixed by the right-hand rule, and it exists only in three dimensions.
The cross product is anticommutative, so swapping the order flips every component's sign. The magnitude is unchanged; only the direction reverses, because you are curling your fingers the opposite way.
The two vectors are parallel or antiparallel, so the angle between them is zero or 180 degrees and its sine is zero. It also happens if either vector has zero length.
Build two edge vectors by subtracting one vertex from the other two, cross them, and halve the magnitude. This works in three dimensions as well as in the plane.
Because the position vector from the pivot must come first. Reversing the order gives the negative, which describes rotation in the opposite direction and will produce the wrong sign throughout a dynamics problem.
It gives the signed volume of the parallelepiped spanned by the three vectors. A value of zero means all three lie in a single plane, which makes it a quick coplanarity test.
Not as a vector. In the plane the usual substitute is the scalar a1b2 minus a2b1, which gives the signed area. In four dimensions and above no vector-valued equivalent exists, and the wedge product is used instead.
