Cross Product Calculator
Calculate the cross product of two 3D vectors — get the perpendicular vector, its magnitude, the angle between vectors, and the parallelogram area.
Results
The Cross Product
The cross product A×B of two 3D vectors produces a new vector that is perpendicular to both A and B. Unlike the dot product (which gives a scalar), the cross product gives a vector. It is defined only in 3D space and is fundamental in physics, computer graphics, and engineering.
The Formula
For A=(ax,ay,az) and B=(bx,by,bz):
A×B = (ay·bz − az·by, az·bx − ax·bz, ax·by − ay·bx)
You can remember this using the determinant of a 3×3 matrix with unit vectors i, j, k in the first row, A components in the second row, and B components in the third row.
Magnitude and Angle
|A×B| = |A||B|sin(θ), where θ is the angle between the vectors. This means the magnitude of the cross product equals the area of the parallelogram formed by A and B. When A and B are parallel (θ=0°), sin(0°)=0, so the cross product is the zero vector. When perpendicular (θ=90°), sin(90°)=1, and the cross product has maximum magnitude.
Right-Hand Rule
The direction of A×B follows the right-hand rule: point your fingers along A, curl them toward B, and your thumb points in the direction of A×B. This means A×B = −(B×A) — swapping the order reverses the direction (anti-commutative property).
Physics Applications
Torque: τ = r×F. The torque from a force F applied at position r is the cross product of the two vectors. Angular momentum: L = r×p (position cross linear momentum). Magnetic force: F = q(v×B) — the force on a moving charge is perpendicular to both velocity and magnetic field.
Frequently Asked Questions
The cross product A×B is a vector perpendicular to both A and B. Its components are (ay·bz−az·by, az·bx−ax·bz, ax·by−ay·bx). Its magnitude equals |A||B|sin(θ), the area of the parallelogram formed by the two vectors.
The vectors are parallel (or one is zero). sin(0°)=0, so A×B=(0,0,0). Example: A=(1,2,3) and B=(2,4,6)=2A are parallel — their cross product is the zero vector.
Point the fingers of your right hand along A, then curl them toward B (through the smaller angle). Your extended thumb points in the direction of A×B. Reversing the order (B×A) points your thumb in the opposite direction.
Torque τ = r×F, where r is the lever arm (position vector from pivot to where force is applied) and F is the force. The torque magnitude is |r||F|sin(θ), maximized when the force is perpendicular to the lever arm (θ=90°).
The cross product as A×B giving a vector perpendicular to both inputs only works in 3D. In 2D, the "cross product" gives a scalar (the z-component of the 3D result). In other dimensions, you need the generalized wedge product from exterior algebra.