Vector Cross Product Calculator

What this vector cross product calculator shows

This vector cross product calculator takes two 3D vectors and returns the perpendicular vector produced by A × B. The sign pattern of the components follows the right-hand rule, while the length of the result tells you how much area the two vectors span in a parallelogram. That combination of direction and magnitude is why the cross product shows up in mechanics, geometry, and 3D graphics.

If you have two vectors A and B in 3D, written in component form as

then the cross product is another vector, written as . In the calculator below, those symbols are the three coordinates you enter for Vector A and Vector B, so the result is tied directly to the numbers on the page rather than to an abstract formula alone.

Cross product formulas used by this calculator

There are two standard ways to write the vector cross product: as a determinant and as explicit component formulas. This page shows both because one is easier to remember by hand and the other is exactly what the calculator evaluates.

Determinant form for the vector cross product

A compact way to remember the cross product is as a 3 × 3 determinant involving the unit vectors i, j, k:

Component formula used by this calculator

Expanding the determinant produces the coordinate-by-coordinate form the calculator uses internally. Each output component blends two inputs from A and two from B, so a small sign change in any entry changes the final vector.

where

The calculator on this page evaluates exactly these formulas in JavaScript, so the numbers shown in the result box match the algebra above. When the input vectors point in nearly the same direction, the cross product shrinks toward zero because the sine term in |A||B|sin(θ) gets small; when the vectors are perpendicular, the magnitude reaches its maximum for those lengths.

How to interpret a vector cross product result

• Direction: The output vector is perpendicular to both input vectors. Use the right-hand rule: point your index finger along A, your middle finger along B, and your thumb indicates the direction of A × B. Because the operation is anti-commutative, swapping the inputs flips that direction.
• Magnitude: The size of the cross product equals |A||B|sin(θ), where θ is the angle between the vectors. That value is the area of the parallelogram spanned by A and B, so if you need the corresponding triangle area, divide by two.
• Zero result: If the cross product is the zero vector, then the original vectors are parallel, anti-parallel, or one of them is the zero vector. In that case the vectors do not define a nonzero area or a unique perpendicular direction.

Worked example: cross product of (1, 2, 3) and (4, 5, 6)

A small numeric example makes the sign pattern easy to check by hand.

• A = (1, 2, 3)
• B = (4, 5, 6)

Using the component formula:

• x-component: cx = ay · bz − az · by = 2 × 6 − 3 × 5 = 12 − 15 = -3
• y-component: cy = az · bx − ax · bz = 3 × 4 − 1 × 6 = 12 − 6 = 6
• z-component: cz = ax · by − ay · bx = 1 × 5 − 2 × 4 = 5 − 8 = -3

So

A × B = (-3, 6, -3).

If you enter these components into the calculator, it returns the same result vector and the matching magnitude. This kind of check is useful when you want to confirm a homework answer, compare a hand calculation with software, or make sure a sign was copied correctly from a diagram.

Cross product vs. dot product in 3D vector math

People often compare the cross product with the dot product because both combine two vectors, but they answer different questions. The cross product gives a perpendicular vector and area information; the dot product gives a scalar that measures alignment. The table below summarizes the key differences.

Common applications of the vector cross product

In practice, the vector cross product is useful anywhere you need a perpendicular direction or an area-like quantity from two 3D directions. The examples below are the most common reasons people reach for this calculator.

• Torque: In mechanics, torque uses a position vector crossed with a force vector to describe rotational tendency around a pivot. The larger the lever arm and the more the force leans away from the line of action, the larger the rotational effect.
• Magnetic (Lorentz) force: In electromagnetism, the force on a charged particle moving in a magnetic field involves the cross product of velocity and magnetic field vectors. The result helps determine the direction that the force pushes the charge through space.
• Surface normals in 3D graphics: Graphics engines use cross products to compute normal vectors to surfaces, which are then used for lighting and shading calculations. A stable normal is essential when a renderer needs to know which side of a polygon faces the camera or a light source.
• Area and orientation: The magnitude of the cross product gives the area of a parallelogram defined by two vectors, while its direction indicates the orientation in 3D space. This is why the same calculation can answer both a geometry question and a direction question at once.

Assumptions and limitations of this vector cross product calculator

This calculator is designed for straightforward numeric 3D cross products, so it follows the standard coordinate formula without symbolic algebra or special cases.

• 3D vectors only: Enter x, y, and z components for each vector. The page does not evaluate 2D vector products or higher-dimensional generalizations directly.
• Numeric inputs: Use integers or decimals, positive or negative. Expressions, variables, and labels are not interpreted as math.
• Floating-point arithmetic: The calculation runs in your browser with standard floating-point arithmetic, so extremely large or tiny values may show rounding effects.
• Ordering matters: A × B is the negative of B × A. If you swap the input vectors, the magnitude stays the same but the direction flips.
• Privacy and offline use: All computation stays in the browser. The values are not sent to a server, and once the page has loaded it continues to work without an internet connection.

Using the vector cross product calculator effectively

When you enter vectors into this calculator, start by checking each coordinate and the order of the vectors.

• Enter the x, y, and z components of Vector A and Vector B into the corresponding fields.
• Use negative signs where needed, and decimals for non-integer values.
• Run the calculation to see the resulting vector components and magnitude.
• If you get a zero vector or a very small result, check whether the inputs are parallel or whether one sign was entered incorrectly.

With the formulas, interpretation tips, and examples above, you can use this vector cross product calculator to verify hand calculations, read the direction of the result correctly, and connect the numbers back to the geometry of the two input vectors.