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Calculator · math

Vector Cross Product Calculator

Compute the cross product A×B of two 3D vectors. Returns the full result vector, scalar magnitude, or individual i, j, k components instantly.

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Inputs

Vector A — i component (a₁)

Vector A — j component (a₂)

Vector A — k component (a₃)

Vector B — i component (b₁)

Vector B — j component (b₂)

Vector B — k component (b₃)

Result to display

Cross Product Result

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Get a plain-English breakdown of your result with practical next steps.

Cross Product Result—

The formula

How the
result is
computed.

What Is the Vector Cross Product?

The cross product is a binary operation on two three-dimensional vectors that produces a third vector perpendicular to both inputs. Defined exclusively for vectors in R³, it is indispensable in physics, engineering, and computer graphics wherever rotational relationships, torques, or surface normals must be computed. Unlike the dot product, which collapses two vectors into a scalar, the cross product yields a full directional vector whose orientation follows the right-hand rule.

The Cross Product Formula

For vector A = (a₁, a₂, a₃) and vector B = (b₁, b₂, b₃), the cross product A × B is computed component by component:

i-component: a₂b₃ − a₃b₂
j-component: a₃b₁ − a₁b₃
k-component: a₁b₂ − a₂b₁

The complete result is: A × B = (a₂b₃ − a₃b₂)î + (a₃b₁ − a₁b₃)ĵ + (a₁b₂ − a₂b₁)k̂. This formula derives from expanding the determinant of a 3×3 matrix whose first row holds the unit vectors î, ĵ, k̂ and whose second and third rows hold the components of A and B, a derivation thoroughly covered by Paul's Online Math Notes — Cross Product.

Understanding Each Variable

Every input maps to a specific spatial dimension:

a₁ — x-axis (i) projection of vector A.
a₂ — y-axis (j) projection of vector A.
a₃ — z-axis (k) projection of vector A.
b₁, b₂, b₃ — the corresponding x-, y-, and z-axis projections of vector B.

The output selector allows display of the full three-component result vector, its scalar magnitude, or any single component (i, j, or k) of A × B — useful when only one directional element is needed for a downstream calculation.

Worked Example

Let A = (1, 2, 3) and B = (4, 5, 6). Applying the formula step by step:

i: (2)(6) − (3)(5) = 12 − 15 = −3
j: (3)(4) − (1)(6) = 12 − 6 = 6
k: (1)(5) − (2)(4) = 5 − 8 = −3

Result: A × B = (−3, 6, −3). The magnitude is √(9 + 36 + 9) = √54 ≈ 7.348. As a perpendicularity check, (−3)(1) + (6)(2) + (−3)(3) = −3 + 12 − 9 = 0, confirming A × B is orthogonal to A.

Magnitude and Geometric Meaning

The scalar magnitude satisfies |A × B| = |A||B|sin(θ), where θ is the angle between A and B. This value equals the area of the parallelogram spanned by A and B — a relationship with direct applications in computational geometry, as documented by Whitman College Calculus, Section 14.4. When θ = 90° the magnitude is maximized; when A and B are parallel (θ = 0°) the cross product is the zero vector.

Real-World Applications

Torque in mechanics: τ = r × F. A 0.3 m wrench applying 50 N perpendicular to the handle produces τ = 15 N·m.
Electromagnetism: The Lorentz force F = qv × B links a charged particle's velocity and an external magnetic field to the resulting force direction.
3D computer graphics: Surface normal vectors, required for lighting and shading, are computed as the cross product of two edge vectors of a polygon.
Structural engineering: Moment of a force about a point is M = r × F, directly applying the cross product in statics and dynamics problems.
Computational geometry: Triangle area equals half the magnitude of the cross product of two edge vectors — a fast method used in mesh processing and GIS.

Key Mathematical Properties

Anti-commutativity: A × B = −(B × A). Order matters: swapping operands negates the result.
Distributivity over addition: A × (B + C) = (A × B) + (A × C).
Scalar associativity: (cA) × B = c(A × B) for any scalar c.
Not associative: (A × B) × C ≠ A × (B × C) in general — a common source of errors in multi-step calculations.
Self-cross-product: A × A = 0 for any vector A.

Reference

Frequently asked questions

What is the cross product of two vectors?

The cross product of two three-dimensional vectors A and B is a new vector perpendicular to both, with direction determined by the right-hand rule. Its magnitude equals |A||B|sin(θ), where θ is the angle between the vectors. The operation is anti-commutative — reversing the operands reverses the result direction — and is defined only in three (and seven) dimensions, distinguishing it from the dot product which works in any dimension.

How do you calculate the cross product step by step?

Given A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃), compute three components: i = a₂b₃ − a₃b₂, j = a₃b₁ − a₁b₃, k = a₁b₂ − a₂b₁. For example, A = (2, 3, 4) and B = (5, 6, 7) gives i = (3)(7)−(4)(6) = −3, j = (4)(5)−(2)(7) = 6, k = (2)(6)−(3)(5) = −3, so A × B = (−3, 6, −3). A cross product calculator automates all three subtractions simultaneously, eliminating sign errors.

Is the cross product commutative?

No — the cross product is anti-commutative, meaning A × B = −(B × A). Swapping the two input vectors produces a result with identical magnitude but opposite direction. This follows from the right-hand rule: curling fingers from A toward B points the thumb in one direction; curling from B toward A points it the opposite way. Confusing the order is one of the most common mistakes in torque and magnetic force calculations, where direction is physically critical.

What does the magnitude of a cross product represent geometrically?

The magnitude |A × B| equals the area of the parallelogram whose two adjacent sides are vectors A and B. The formula |A × B| = |A||B|sin(θ) shows this directly. For |A| = 6, |B| = 4, and θ = 45°, the parallelogram area is 6 × 4 × 0.7071 ≈ 16.97 square units. Dividing by 2 gives the area of the triangle formed by A and B, a shortcut widely used in mesh processing, finite element analysis, and geographic information systems.

What are real-world applications of a cross product calculator?

A cross product calculator directly supports torque computation (τ = r × F) in mechanical engineering, magnetic force determination (F = qv × B) in electrophysics, surface normal generation in 3D rendering engines, angular momentum analysis in rotational dynamics, and polygon area computation in computational geometry. Structural engineers use it for moment-of-force problems; game developers use it to orient camera and lighting vectors; robotics engineers use it to compute joint axes. Automating the arithmetic eliminates sign errors that commonly occur in the j-component calculation.

What is the difference between the dot product and the cross product?

The dot product A · B = |A||B|cos(θ) produces a scalar measuring directional alignment — it is commutative and defined in any number of dimensions. The cross product A × B = |A||B|sin(θ) produces a perpendicular vector measuring the rotational or area relationship — it is anti-commutative and only defined in three dimensions. Use the dot product for work, projection, and angle-between-vectors problems; use the cross product for torque, normal vectors, and parallelogram area problems. Both operations together fully characterize the geometric relationship between two 3D vectors.