Cross Product Calculator

Compute the vector cross product A × B for 3D vectors with magnitude, direction, and step-by-step solution.

Enter Two 3D Vectors

Vector A = (a1, a2, a3)
Vector B = (b1, b2, b3)

Result

A × B
(-3, 6, -3)
cross product vector
i component-3
j component6
k component-3
Magnitude |A x B|--
|A|--
|B|--
Angle between A and B--
Parallelogram area--

Step-by-Step Solution

A x B = |i j k; a1 a2 a3; b1 b2 b3|

Understanding the Cross Product

The cross product (also called the vector product) is a binary operation on two vectors in three-dimensional space. Unlike the dot product, which produces a scalar, the cross product produces a new vector that is perpendicular to both input vectors. The cross product is fundamental in physics, engineering, and computer graphics.

Given two vectors A = (a1, a2, a3) and B = (b1, b2, b3), their cross product A × B is computed using a determinant formula involving the unit vectors i, j, and k. The resulting vector is orthogonal to both A and B, and its magnitude equals the area of the parallelogram formed by A and B.

Cross Product Formulas

Determinant Formula

The cross product is computed via a 3x3 determinant with unit vectors in the first row.

A x B = |i j k; a1 a2 a3; b1 b2 b3|

Component Form

Expanding the determinant gives three components.

A x B = (a2*b3 - a3*b2)i - (a1*b3 - a3*b1)j + (a1*b2 - a2*b1)k

Magnitude

The magnitude of the cross product relates to the sine of the angle between vectors.

|A x B| = |A| * |B| * sin(theta)

Parallelogram Area

The magnitude of the cross product equals the area of the parallelogram spanned by A and B.

Area = |A x B|

Anti-Commutativity

The cross product is anti-commutative: reversing the order negates the result.

B x A = -(A x B)

Right-Hand Rule

The direction follows the right-hand rule: curl fingers from A to B, thumb points along A x B.

Direction: perpendicular to both A and B

How to Compute the Cross Product

Step-by-Step Process

  1. Write the two vectors as A = (a1, a2, a3) and B = (b1, b2, b3).
  2. Set up the 3x3 determinant: first row is i, j, k; second row is A; third row is B.
  3. Compute the i-component: a2*b3 - a3*b2.
  4. Compute the j-component: -(a1*b3 - a3*b1) = a3*b1 - a1*b3.
  5. Compute the k-component: a1*b2 - a2*b1.
  6. Combine: A x B = (i-comp, j-comp, k-comp).
  7. Calculate the magnitude: sqrt(i^2 + j^2 + k^2).

Properties of the Cross Product

  • Anti-commutative: A x B = -(B x A).
  • Distributive: A x (B + C) = A x B + A x C.
  • Scalar multiplication: (cA) x B = c(A x B).
  • Parallel vectors: If A is parallel to B, then A x B = 0.
  • Perpendicularity: The result is always perpendicular to both input vectors.
  • Not associative: A x (B x C) is generally not equal to (A x B) x C.

Applications of the Cross Product

The cross product is used extensively in physics for torque (r x F), angular momentum (r x p), and magnetic force (qv x B). In computer graphics, it determines surface normals, performs lighting calculations, and tests polygon winding order. In engineering, it helps analyze rotational forces, moments, and structural loads.

Triangle Area Using Cross Product

The area of a triangle with vertices P, Q, R can be found as half the magnitude of the cross product of two edge vectors: Area = (1/2)|PQ x PR|. This is a powerful computational technique used in 3D geometry and mesh processing.