Vector Cross Product Formula
Calculate the cross product of two 3D vectors.
Find the perpendicular vector, its magnitude, and direction using the right-hand rule.
The Formula
The cross product (also called the vector product) takes two vectors in three-dimensional space and produces a third vector that is perpendicular to both input vectors. This operation is fundamental in physics, engineering, and computer graphics for calculating surface normals, torques, and areas.
Unlike the dot product, which gives a scalar (single number), the cross product gives a full vector with both magnitude and direction. The magnitude of the resulting vector equals the area of the parallelogram formed by the two input vectors. The direction follows the right-hand rule: if you point your right hand's fingers along the first vector and curl them toward the second vector, your thumb points in the direction of the cross product.
The cross product has several important properties. It is anti-commutative, meaning a × b = −(b × a). If two vectors are parallel, their cross product is the zero vector, because no unique perpendicular direction exists. The cross product is also not associative: a × (b × c) does not generally equal (a × b) × c.
The formula can also be expressed as a determinant of a 3×3 matrix with the unit vectors i, j, k in the first row, the components of vector a in the second row, and the components of vector b in the third row. This determinant form is often easier to remember and compute by hand.
Hermann Grassmann and William Rowan Hamilton independently developed the cross product concept in the mid-1800s. Josiah Willard Gibbs later popularized the modern vector notation used today.
Variables
| Symbol | Meaning |
|---|---|
| a = (a1, a2, a3) | First 3D vector |
| b = (b1, b2, b3) | Second 3D vector |
| a × b | Cross product vector (perpendicular to both a and b) |
| |a × b| | Magnitude of cross product = area of parallelogram |
Example 1
Problem
Find the cross product of a = (2, 3, 4) and b = (5, 6, 7).
x-component: (3)(7) − (4)(6) = 21 − 24 = −3
y-component: (4)(5) − (2)(7) = 20 − 14 = 6
z-component: (2)(6) − (3)(5) = 12 − 15 = −3
a × b = (−3, 6, −3). This vector is perpendicular to both a and b.
Example 2
Problem
Find the area of the triangle with vertices at A(1, 0, 0), B(0, 2, 0), and C(0, 0, 3).
AB = B − A = (−1, 2, 0), AC = C − A = (−1, 0, 3)
AB × AC = ((2)(3) − (0)(0), (0)(−1) − (−1)(3), (−1)(0) − (2)(−1))
AB × AC = (6, 3, 2)
|AB × AC| = √(36 + 9 + 4) = √49 = 7
Triangle area = |AB × AC| / 2 = 7/2 = 3.5 square units.
When to Use It
The cross product is essential in any application involving perpendicular directions or areas in 3D space.
- Finding surface normals for lighting calculations in computer graphics and 3D rendering
- Calculating torque in physics (τ = r × F, where r is the position vector and F is the force)
- Determining the area of triangles and parallelograms in 3D space
- Checking if two vectors are parallel (cross product equals zero)