Cross Product Calculator

1. Enter the three components of Vector A.
2. Enter the three components of Vector B.
3. Read the cross product, a new vector.
4. See its magnitude, which is the area of the parallelogram they span.

Here is what each result means:

The cross product, or vector product, multiplies two three-dimensional vectors to produce a third vector. Unlike the dot product, which gives a single number, the cross product gives a vector that points at right angles to both of the originals. Its direction follows the right-hand rule, and its length equals the area of the parallelogram the two vectors form.

The cross product is defined only in three dimensions. It is central to physics and engineering, where it gives torque, angular momentum and magnetic force, and to graphics, where it finds the normal direction of a surface. When two vectors are parallel, their cross product is the zero vector, since they span no area.

The magnitude of the cross product equals the product of the two lengths times the sine of the angle between them. That makes it largest when the vectors are perpendicular and zero when they line up, the mirror image of the dot product, which is largest when they align. Together the two products describe how any pair of vectors relate.

It combines the components in a fixed pattern that produces a perpendicular vector.

1. Form each component from a pair of cross-multiplied terms.
2. Subtract to get the x, y and z parts of the result.
3. Take its length for the area of the parallelogram.

For the scalar product that measures alignment instead, use the dot product calculator.

Find the cross product of 1, 0, 0 and 0, 1, 0.

Calculation: these are the x and y unit vectors. Their cross product is (0, 0, 1), the z unit vector, pointing straight up. Its magnitude is 1, the area of the unit square the two vectors span.

The direction of the cross product follows a simple hand rule.

Because the order flips the direction, A cross B is the negative of B cross A. The cross product is not commutative.

The two vector products answer different questions.

The dot product measures how aligned two vectors are; the cross product measures how much they span, and points out of their plane.

The Angle Between the Vectors

The magnitude is largest when the vectors are perpendicular and zero when they are parallel.

The Order of the Vectors

Swapping A and B reverses the direction of the result, though the magnitude stays the same.

The Lengths of the Vectors

Longer vectors span a larger parallelogram, so the magnitude grows with both lengths.

Physics

Compute torque, angular momentum or the force on a moving charge.

Geometry

Find a vector perpendicular to two others, or the area of a parallelogram or triangle.

Graphics

Get the normal direction of a surface for lighting and orientation.

1. Expecting a Number

The cross product is a vector, not a scalar. The dot product gives a number.

2. Using 2D Vectors

The cross product is defined in three dimensions. Enter three components for each vector.

3. Reversing the Order

A cross B and B cross A point opposite ways. Keep the order you intend.

4. Sign Slips in the Components

Each component subtracts one product from another. A single sign error changes the result.

5. Ignoring the Parallel Case

Parallel vectors give the zero vector, which is correct, not a mistake.

The cross product is exact; only the displayed decimals are rounded.

It is a vector found by multiplying two 3D vectors in a fixed pattern. The result points at right angles to both inputs, and its length equals the area of the parallelogram they span.

Combine the components: the x part is a2 b3 minus a3 b2, and the y and z parts follow the same crossed pattern. This tool computes all three and the magnitude for you.

The dot product gives a single number that measures alignment. The cross product gives a vector perpendicular to both inputs, whose length measures the area they span.

The standard cross product is defined only in three dimensions, where a unique perpendicular direction exists. In two dimensions there is no such vector, and higher dimensions need other tools.

It means the two vectors are parallel or one is zero. Parallel vectors span no area, so their cross product is the zero vector.

It gives the direction of the cross product. Point your right-hand fingers along the first vector, curl them toward the second, and your thumb points along the result.

No. A cross B is the negative of B cross A. Swapping the order keeps the magnitude the same but reverses the direction.

It gives torque and angular momentum in physics, the force on a moving charge, the normal direction of a surface in graphics, and the area of parallelograms and triangles.

No. The calculation runs in your browser and nothing you enter is stored or sent anywhere, unless you choose Save, which keeps the result only on this device.

• Cross product (Wikipedia).
• The cross product explained (Maths Is Fun).
• Right-hand rule (Wikipedia).