Cross Product Calculator

What the cross product computes

The cross product of two three-dimensional vectors a and b is written a x b, and the result is a third vector, not a number. The new vector stands perpendicular to both inputs, its length equals |a| |b| sin(theta), where theta is the angle between them, and its direction follows the right-hand rule: curl the fingers of your right hand from a toward b, and the thumb gives the direction of a x b. Section 1.6 of the Digital Library of Mathematical Functions (NIST, current online edition) lists the component formula and the right-hand rule as equation 1.6.9, and that is the convention this calculator follows.

The cross product and the dot product answer different questions. The dot product returns one number that measures how much the two vectors point the same way. The cross product returns a vector that measures how much the pair fails to line up: when the inputs are parallel, sin(theta) is 0 and the product collapses to the zero vector, and when they are perpendicular, sin(theta) is 1 and the result is as long as it can be, |a| |b|.

Because the output carries a direction, order matters. Swapping the inputs flips the sign: b x a = -(a x b). This anticommutative property is the arithmetic version of the right-hand rule, and it is why cross products appear wherever orientation carries meaning, from torque in mechanics to lighting normals in 3D graphics.

The component formula, worked twice

For a = (a1, a2, a3) and b = (b1, b2, b3), the cross product is a x b = (a2 b3 - a3 b2, a3 b1 - a1 b3, a1 b2 - a2 b1). Each output component pairs the two inputs it does not use, with a sign pattern that cycles forward.

First example, a = (3, 4, 0) and b = (4, 0, 0): c1 = 4 x 0 - 0 x 0 = 0, c2 = 0 x 4 - 3 x 0 = 0, and c3 = 3 x 0 - 4 x 4 = -16, so a x b = (0, 0, -16). The result points down the negative z axis, which the right-hand rule confirms: fingers from (3, 4, 0) toward (4, 0, 0), thumb below the plane.

Second example, a = (1, 2, 3) and b = (4, 5, 6), component by component:

So (1, 2, 3) x (4, 5, 6), which gives (-3, 6, -3). Both results come straight out of the determinant pattern, and the widget on this page runs the same three expressions, printing the components to three decimals.

Magnitude as area, and the angle

The length of the cross product has a geometric reading: |a x b| = |a| |b| sin(theta) is the area of the parallelogram the two vectors span, and half of it is the area of the triangle they span with the origin. This is often the fastest route to an area from coordinates, with no side lengths to work out first.

For a = (3, 4, 0) and b = (4, 0, 0): |a| = 5, |b| = 4, and the dot product a . b = 3 x 4 + 4 x 0 + 0 x 0 = 12. So cos(theta) = 12 / 20 = 0.6, sin(theta) = 16 / 20 = 0.8, and the angle is atan2(16, 12) = 53.13 degrees. The magnitude checks out three ways: directly as sqrt(0 + 0 + 256) = 16, by the sine form as 5 x 4 x 0.8 = 16, and as the parallelogram area, 16.

For (1, 2, 3) and (4, 5, 6): |a| = 3.742, |b| = 8.775, the dot product is 32, the cross product magnitude is 7.348, so sin(theta) = 0.224 and the angle is 12.93 degrees. The parallelogram area is 7.348 and the triangle area is 3.674.

Type these pairs into the widget and it prints the same magnitudes and angles, rounded to the same decimals, together with the two areas.

Parallel, antiparallel and the zero test

A zero cross product is the clean test for parallel vectors. If a x b = (0, 0, 0), the pair is parallel, antiparallel, or one of them is the zero vector, and no unique normal direction exists.

Take a = (2, 1, 0) and b = (4, 2, 0). Here b = 2a, so: c1 = 1 x 0 - 0 x 2 = 0, c2 = 0 x 4 - 2 x 0 = 0, and c3 = 2 x 2 - 1 x 4 = 0. Every component cancels, and the calculator reports magnitude 0, an angle of 0 degrees, and no unit normal rather than a division by zero.

Antiparallel inputs behave the same way. (1, 0, 0) and (-2, 0, 0) also return the zero vector, and the angle reads 180 degrees. The widget states the parallel case in its conclusion line so the empty normal does not look like a bug.

The sign flip cuts the other way too. For a = (3, 4, 0) and b = (4, 0, 0) the product was (0, 0, -16); entering b first gives (0, 0, 16). Same length, opposite direction, exactly what b x a = -(a x b) requires.

Unit normal and which way it points

Dividing the cross product by its own length gives the unit normal n = (a x b) / |a x b|, a length-one vector perpendicular to both inputs. For the pair (3, 4, 0) and (4, 0, 0) it is (0, 0, -1), and for (1, 2, 3) and (4, 5, 6) it is (-0.408, 0.816, -0.408).

The direction is a convention, fixed by the right-hand rule, and it flips when the inputs swap order. If the surface you are working with needs a normal pointing the other way, reverse the argument order; nothing else changes.

The normal earns its keep in 3D work. A game engine takes the normal of each triangle to decide how light strikes it, a physics engine uses normals to resolve collisions, and mesh tools compare the normal they compute with the normal a model claims, which is how flipped or degenerate faces get caught. In all three cases the computation is the same one this widget performs.

Two dimensions: the scalar cross product

In two dimensions there is no third axis for the result to point along, so the cross product is written as a single number: a x b = a1 b2 - a2 b1. Switch the widget to 2D mode and it computes this scalar.

For a = (3, 1) and b = (1, 2): 3 x 2 - 1 x 1 = 5. The magnitude is 5, the angle is atan2(5, 5) = 45 degrees, the parallelogram area is 5 and the triangle area is 2.5. A negative result is information, not an error: it says b sits clockwise from a, which is how 2D orientation tests work.

Summing these scalar products along the edges of a polygon and halving the total gives the polygon area, the shoelace method. The same one-line formula drives survey plots, SVG path tools and the signed area readout in the widget.

Where the number earns its keep

Torque is the everyday cross product. A mechanic pulls a wrench with 80 N at the end of a 0.3 m handle, perpendicular to it: torque = r x F, so the magnitude is 0.3 x 80 x sin 90 = 24 N.m. Pull at 60 degrees instead and the same force delivers 0.3 x 80 x 0.866 = 20.78 N.m, about 13 percent less, which is why grip angle matters as much as grip strength.

Three points P, Q, R define two edge vectors, Q - P and R - P, and half the magnitude of their cross product is the triangle area. The pair (3, 4, 0) and (4, 0, 0) from the first example gives a triangle area of 8; the same three lines of arithmetic area any lot, sail or roof face from coordinates.

Know the edges of the tool: the cross product as defined here exists for 3D vectors (and, as a mathematical curiosity, in 7 dimensions), while the 2D result is a scalar by convention. It is not associative, so (a x b) x c differs from a x (b x c) in general, and it is undefined when either input is the zero vector. MIT's 18.02 multivariable calculus course on OpenCourseWare covers the geometry in its first-week notes, and MathWorld's cross product entry collects the identities, including the quaternion and determinant forms.

To use the widget: pick 3D or 2D, type the components, and read the product, magnitude, angle, unit normal and areas off one table. Every value updates as you type, so you can test a pair, swap the order, and watch the sign flip.