Cross Product Calculator

What this calculator does

The cross product is the other way to multiply two vectors, and unlike the dot product it gives back a vector, not a number. That new vector points at right angles to both of the originals. This computes it for two 3D vectors.

Enter vector a and vector b and it returns their cross product. It runs right here in the browser.

Using the calculator

1. Enter the x, y and z components of vector a.
2. Enter the x, y and z components of vector b.
3. Press Calculate.

It returns the cross product as a vector with three components. Reset clears the boxes.

What the cross product is

Give the cross product two vectors and it hands you a third one that is perpendicular to both, pointing straight out of the flat sheet the first two lie in. For that reason it is also called the vector product, to mark the contrast with the dot product's single number.

It lives only in three dimensions. That is not a shortcoming, it is the whole idea: the cross product's job is to point out of a plane, and only in 3D is there a direction to point into. In 2D there is no room for it, which is why the tool asks for a full x, y and z on each vector.

The formula

For a = (a_x, a_y, a_z) and b = (b_x, b_y, b_z), the cross product is:

a × b = ( a_yb_z − a_zb_y,  a_zb_x − a_xb_z,  a_xb_y − a_yb_x )

Each component is a small criss-cross of the other two axes, which is where the name and the × symbol come from. It is really the expansion of a 3 by 3 determinant, and the pattern of which terms subtract from which is exactly what makes the result come out perpendicular to both vectors. The tool follows this formula precisely.

Which way it points: the right-hand rule

Being perpendicular to the plane leaves two possible directions, straight up or straight down out of it. The one the cross product picks is given by the right-hand rule: point the fingers of your right hand along vector a, curl them toward vector b, and your thumb points along a × b.

A consequence is that order matters. Swap the two vectors and the result flips to point the opposite way, so b × a is the exact negative of a × b. This is unlike the dot product, where order makes no difference at all.

Its length is an area

The direction of the cross product is one half of the story; its length is the other. The magnitude of a × b equals |a| × |b| × sin(θ), which happens to be the area of the parallelogram that the two vectors frame. That leads to a neat companion to the dot product's perpendicular test: because sin(0) is zero, two parallel vectors have a cross product of zero. So a zero cross product is the test for two vectors pointing along the same line, just as a zero dot product is the test for a right angle.

A worked example

Cross the x-axis with the y-axis: a = (1, 0, 0) and b = (0, 1, 0).

• First component: a_yb_z − a_zb_y = 0 × 0 − 0 × 1 = 0.
• Second component: a_zb_x − a_xb_z = 0 × 0 − 1 × 0 = 0.
• Third component: a_xb_y − a_yb_x = 1 × 1 − 0 × 0 = 1.

So a × b = (0, 0, 1), the z-axis, standing straight up perpendicular to the flat xy-plane the other two lie in. Just as the rule promises, b × a would give (0, 0, -1), pointing the opposite way.

What it is used for

The cross product turns up wherever a perpendicular direction or a rotation is involved. In computer graphics it finds the normal to a surface, the direction it faces, needed for lighting. In physics it defines torque, the twisting effect of a force, and other rotational quantities. Its length gives the area of a triangle or parallelogram from its edge vectors. And it offers a quick parallel test, since a cross product of zero means the vectors line up. Its partner is the dot product, which measures alignment instead of building a perpendicular.

Questions people ask

What is the cross product?

A way of multiplying two 3D vectors to get a new vector that is perpendicular to both. It is also called the vector product.

Is the result a vector or a number?

A vector. This is the key difference from the dot product, which gives a single number.

Why does the cross product only work in 3D?

Its result points out of the plane of the two vectors, and only three dimensions provide a direction for it to point into. There is no room for it in 2D.

Does the order of the vectors matter?

Yes. Swapping them reverses the result, so b × a is the negative of a × b. The right-hand rule tells you which way each points.

How do I check if two vectors are parallel?

Take their cross product. If it is the zero vector, they point along the same line.

References

A note on where this comes from. The cross product, or vector product, multiplies two vectors in three dimensions to give a vector perpendicular to both, with a length equal to the area of the parallelogram they span. The name "vector product" was coined by William Kingdon Clifford in 1877 to stress that its result is a vector, and the modern × notation is due to Gibbs and Heaviside. For further reading, see Cross product.

1. The cross product, giving a vector perpendicular to both inputs, with magnitude |a||b|sin(θ) equal to the parallelogram area.
2. W. K. Clifford, who named it the vector product (1877) to emphasise that the result is a vector, not a scalar.