Multiply two vectors together and you might expect a number. The cross product does something stranger and more useful: it produces a third vector, standing at right angles to both of the originals. That perpendicular direction is exactly what physics and 3D graphics constantly need β the axis a torque acts around, the direction a charged particle is pushed by a magnetic field, the way a surface faces in a rendered scene. The cross product calculator computes this vector product for two three-dimensional vectors, returning the components of the result so you can use them directly.
Quick Answer
For a = (aβ, aβ, aβ) and b = (bβ, bβ, bβ), the cross product is a Γ b = (aβbβ β aβbβ, aβbβ β aβbβ, aβbβ β aβbβ). The middle component uses the pattern in reverse, which is where its minus sign comes from β the detail most often dropped when working by hand.
A Vector That Points Out of the Plane
Any two vectors that are not parallel define a flat plane. The cross product points straight out of that plane, perpendicular to both, and its length carries geometric meaning: |a Γ b| equals |a| |b| sin ΞΈ, which is exactly the area of the parallelogram the two vectors span. Double either vector and the area, and the cross product, doubles too.
Direction is fixed by the right-hand rule: point the fingers of your right hand along the first vector, curl them toward the second, and your thumb points along a Γ b. The unit vectors obey the tidy cycle i Γ j = k, j Γ k = i, k Γ i = j. Because the result is perpendicular to both inputs, it is automatically a normal vector to the plane they define β the reason graphics engines use cross products to find which way a polygon faces and whether it catches the light.
A built-in accuracy check comes free: the dot product of a Γ b with either a or b must be zero, since perpendicular vectors have a dot product of zero. If your hand calculation fails that test, an arithmetic slip is hiding somewhere.
By hand, the formula is usually organised as a determinant with the unit vectors across the top row β i, j, k in the first row, a's components in the second, b's in the third β and expanded along that top row. The middle term picks up a minus sign in the expansion, which is precisely why the second component of the result is written aβbβ β aβbβ rather than the other way around. Keeping that sign straight is the entire battle; the calculator removes it, but knowing where it comes from helps you spot implausible answers.
The applications explain why anyone bothers. Torque on a wrench is Ο = r Γ F: the force does its turning work only through the component perpendicular to the handle, and the cross product captures exactly that, direction of the axle included. A charged particle moving through a magnetic field feels a force proportional to v Γ B, deflected sideways to both its motion and the field β the effect that steers beams in particle accelerators and bends the paths that produce auroras. Angular momentum, the Coriolis effect and the surface normals of 3D rendering all run through the same operation.
How to Use the Cross Product Calculator
- Write your first vector's components in order: x (i), y (j), z (k).
- Enter those three numbers as vector a.
- Write the second vector's components in the same order and enter them as vector b.
- Double-check the order of the two vectors β swapping them reverses the answer's direction, as explained below.
- Run the calculation and read the three components of the resulting vector.
- If you need the magnitude as well, it is the square root of the sum of the squared components of the result.
Worked Example: (2, 3, 4) Γ (5, 6, 7)
Let a = (2, 3, 4) and b = (5, 6, 7). Apply the formula component by component.
First component: aβbβ β aβbβ = (3)(7) β (4)(6) = 21 β 24 = β3. Second component: aβbβ β aβbβ = (4)(5) β (2)(7) = 20 β 14 = 6. Third component: aβbβ β aβbβ = (2)(6) β (3)(5) = 12 β 15 = β3.
So a Γ b = (β3, 6, β3). Its magnitude is β(9 + 36 + 9) = β54 β 7.35, which is also the area of the parallelogram spanned by a and b. Now verify the perpendicularity: the dot product with a is (β3)(2) + (6)(3) + (β3)(4) = β6 + 18 β 12 = 0, and with b it is (β3)(5) + (6)(6) + (β3)(7) = β15 + 36 β 21 = 0. Zero both times β the result is genuinely perpendicular to both inputs, so the arithmetic is consistent.
Order Matters: a Γ b Is Not b Γ a
The cross product is anticommutative: b Γ a = β(a Γ b), every component flipped in sign. In the example above, reversing the order gives (3, β6, 3) β same line, opposite direction. Whenever direction carries meaning, as with torque or rotation axes, the order must match the physical setup. Other traps: parallel or identical vectors give the zero vector, because sin 0Β° = 0, which is a useful test for parallelism rather than an error. The operation is defined for three-dimensional vectors; for a 2D problem, give both vectors a z-component of 0 and the result points purely along z. And do not confuse the cross product with the dot product β one returns a vector, the other a single number, and substituting one for the other wrecks everything downstream.
More Tools for Vector and Geometry Work
The magnitude calculations behind this tool are three-dimensional Pythagoras, and the Pythagorean Theorem Calculator covers the flat version in depth. For the distance between the tips of two position vectors, the Distance Formula Calculator is the natural companion. Both live in the math section with the rest of the geometry tools.