Vector Cross Product Calculator

The cross product (also called the vector product) takes two vectors in three-dimensional space and produces a third vector perpendicular to both. Unlike the dot product, which collapses two vectors into a single number, the cross product preserves direction. The result is a full vector with magnitude and orientation.

The formula:

a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)

Or as a 3×3 determinant with the standard basis vectors in the top row:

| i j k | | a₁ a₂ a₃ | | b₁ b₂ b₃ |

Expanding along the top row gives the same component expression. Many people find the determinant form easier to remember and harder to make sign errors with.

The magnitude has a clean geometric meaning:

|a × b| = |a| × |b| × sin(θ)

Where θ is the angle between the two vectors. This equals the area of the parallelogram with sides a and b. Half that value is the area of the triangle with those sides as edges.

Right-hand rule for direction: Point your right hand’s fingers along vector a, curl them toward vector b through the smaller angle, and your thumb points in the direction of a × b. If a and b are parallel, no unique perpendicular exists and the cross product is the zero vector. If they are antiparallel, same thing.

Key properties:

• Anti-commutative: a × b = −(b × a). Order matters.
• Distributive: a × (b + c) = a × b + a × c
• Not associative: a × (b × c) ≠ (a × b) × c in general
• a × a = 0 for any vector a
• Scalar pull-through: (k·a) × b = k·(a × b)

Worked example, a × b for a = (2, 3, 4) and b = (5, 6, 7):

x-component: a₂b₃ − a₃b₂ = (3)(7) − (4)(6) = 21 − 24 = −3 y-component: a₃b₁ − a₁b₃ = (4)(5) − (2)(7) = 20 − 14 = 6 z-component: a₁b₂ − a₂b₁ = (2)(6) − (3)(5) = 12 − 15 = −3

a × b = (−3, 6, −3)

Check perpendicularity: a · (a × b) = (2)(−3) + (3)(6) + (4)(−3) = −6 + 18 − 12 = 0 ✓ Similarly b · (a × b) = (5)(−3) + (6)(6) + (7)(−3) = −15 + 36 − 21 = 0 ✓

Worked example, area of a triangle in 3D: Triangle with vertices A(1, 0, 0), B(0, 2, 0), 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)) = (6, 3, 2) |AB × AC| = √(36 + 9 + 4) = √49 = 7 Triangle area = |AB × AC| / 2 = 3.5 square units

Where the cross product shows up:

• Physics torque: τ = r × F. The torque magnitude equals |r||F|sin(θ), which is why a force perpendicular to the lever arm produces maximum torque.
• Magnetic force on a moving charge: F = qv × B. This is why charged particles spiral in magnetic fields rather than going straight.
• Angular momentum: L = r × p, where p is linear momentum. Conservation of angular momentum follows directly from this definition.
• Computer graphics: surface normals are computed as the cross product of two edge vectors of a triangle. The normal direction determines which side of the triangle faces the camera and how light reflects off it.
• Aerospace and robotics: orientation calculations, rigid-body rotations, and inertia tensor work all rely heavily on cross products.

Historical note: Hermann Grassmann published the geometric foundations in 1844. William Rowan Hamilton developed quaternions in 1843, and the cross product fell out naturally as the imaginary-part operation. Josiah Willard Gibbs and Oliver Heaviside in the 1880s extracted the modern vector-only notation from Hamilton’s quaternion algebra, which is what most physics and engineering students learn today.

Cross product vs. dot product: The dot product a · b = |a||b|cos(θ) gives a scalar that measures alignment. The cross product gives a vector that measures perpendicularity. Together they characterize the relationship between two vectors completely: if you know |a|, |b|, the dot product, and the cross product, the geometry is fully determined.