Vector Cross Product Calculator
Compute the cross product of two 3D vectors A and B, returning the resulting perpendicular vector. Ideal for physics, engineering, and graphics problems involving torque, normals, or rotational axes.
About this calculator
The cross product A × B of two 3D vectors produces a new vector perpendicular to both. Given A = (a_x, a_y, a_z) and B = (b_x, b_y, b_z), the result C = (c_x, c_y, c_z) is computed component-wise using the determinant of a symbolic 3×3 matrix: c_x = a_y·b_z − a_z·b_y, c_y = a_z·b_x − a_x·b_z, c_z = a_x·b_y − a_y·b_x. The magnitude |A × B| = |A||B|sin(θ), where θ is the angle between the vectors. This magnitude equals the area of the parallelogram spanned by A and B. If A × B = 0, the vectors are parallel or one is zero. The direction of the result follows the right-hand rule.
How to use
Let A = (1, 2, 3) and B = (4, 5, 6). Enter a_x=1, a_y=2, a_z=3, b_x=4, b_y=5, b_z=6. Compute each component: c_x = a_y·b_z − a_z·b_y = 2·6 − 3·5 = 12 − 15 = −3. c_y = a_z·b_x − a_x·b_z = 3·4 − 1·6 = 12 − 6 = 6. c_z = a_x·b_y − a_y·b_x = 1·5 − 2·4 = 5 − 8 = −3. The cross product is (−3, 6, −3). You can verify this is perpendicular to both A and B by checking that the dot products A·C and B·C both equal zero.
Frequently asked questions
What is the geometric meaning of the cross product of two vectors?
Geometrically, the cross product A × B yields a vector orthogonal to the plane containing A and B. Its magnitude equals the area of the parallelogram formed by the two vectors as adjacent sides. The direction is determined by the right-hand rule: if you curl the fingers of your right hand from A toward B, your thumb points in the direction of A × B. This makes it invaluable for computing surface normals in 3D graphics and torques in physics.
How does the cross product differ from the dot product?
The dot product of two vectors returns a scalar and measures how much the vectors align (A·B = |A||B|cos θ). The cross product returns a vector perpendicular to both inputs and measures how much they differ in direction (|A×B| = |A||B|sin θ). The dot product is commutative (A·B = B·A), while the cross product is anti-commutative: A×B = −(B×A). These two operations complement each other in 3D vector analysis.
When is the cross product used in physics and engineering applications?
The cross product appears wherever rotation or perpendicularity matters. Torque is calculated as τ = r × F, where r is the position vector and F is the force. Angular momentum is L = r × p. In electromagnetism, the magnetic force on a moving charge is F = q(v × B). In computer graphics, cross products generate surface normal vectors needed for lighting calculations and collision detection.