Cross Product Calculator
Calculate the cross product of two 3D vectors, with the resulting vector and its magnitude.
➕ What is the Cross Product Calculator?
The cross product calculator finds the vector cross product of two 3D vectors, producing a new vector that is perpendicular to both original vectors, along with its magnitude.
Students studying vector algebra and physics use cross product for calculating torque, angular momentum, and magnetic force. Engineers and 3D graphics programmers use it to find surface normals, determine rotational direction, and compute areas of parallelograms and triangles in 3D space.
A common point of confusion is mixing up cross product with dot product. The dot product produces a single number (scalar) describing how aligned two vectors are. The cross product instead produces an entirely new vector, perpendicular to both inputs, with a magnitude related to the area they sweep out together.
This tool is useful because it shows the full component-by-component working for each of the three resulting vector components, alongside the final magnitude, so you can verify every step of the calculation.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - A=(1,2,3), B=(4,5,6)
Example 2 - A=(2,0,0), B=(0,3,0)
Example 3 - Unit vectors A=(1,0,0), B=(0,1,0)
❓ Frequently Asked Questions
🔗 Related Calculators
How do you calculate the cross product of two vectors?
For vectors A = (Ax, Ay, Az) and B = (Bx, By, Bz), the cross product is (AyBz - AzBy, AzBx - AxBz, AxBy - AyBx). For A = (1,2,3) and B = (4,5,6), the cross product is (2x6-3x5, 3x4-1x6, 1x5-2x4) = (-3, 6, -3).
What does the cross product represent geometrically?
The cross product of two vectors produces a third vector that is perpendicular to both original vectors, with a direction determined by the right-hand rule. Its magnitude equals the area of the parallelogram formed by the two original vectors.
What is the difference between cross product and dot product?
The dot product of two vectors produces a scalar (single number) representing how much the vectors point in the same direction. The cross product produces a new vector perpendicular to both inputs, with a magnitude related to the sine of the angle between them rather than the cosine.
Does the order of vectors matter in a cross product?
Yes. Cross product is anti-commutative: A cross B equals the negative of B cross A. Swapping the order of the two vectors reverses the direction of the resulting vector (flips the sign of every component) but keeps the same magnitude.
What does it mean if the cross product is the zero vector?
A cross product of (0, 0, 0) means the two vectors are parallel (or anti-parallel), including the case where one or both vectors are the zero vector. Parallel vectors do not define a unique perpendicular direction, so their cross product has no meaningful direction and zero magnitude.
How is cross product used in physics?
Cross product appears throughout physics: torque equals the cross product of a force and a lever-arm vector, angular momentum equals the cross product of position and linear momentum, and magnetic force on a moving charge equals the cross product of velocity and magnetic field (scaled by charge).
How is the right-hand rule used to find cross product direction?
Point the fingers of your right hand in the direction of the first vector, then curl them toward the second vector; your thumb points in the direction of the cross product. This gives the same direction the mathematical formula produces, and is a useful way to sanity-check a calculated result.
Can you take the cross product of 2D vectors?
The cross product is formally defined for 3D vectors. For 2D vectors, treat them as 3D vectors with a z-component of zero; the result will have zero x and y components, and the z component gives a scalar equivalent commonly used to determine 2D rotational direction (clockwise versus counterclockwise).
How is cross product magnitude related to the area of a parallelogram?
The magnitude of A cross B equals the area of the parallelogram formed by placing vectors A and B tail to tail. This follows from the formula |A||B|sin(theta), where sin(theta) accounts for the effective 'height' of the parallelogram relative to the base vector.
What is the cross product of a vector with itself?
The cross product of any vector with itself is always the zero vector (0, 0, 0), since a vector is always parallel to itself, and the sine of the angle between two parallel vectors (0 degrees) is zero.