Cross Product Calculator

What Is the Cross Product?

The cross product u × v of two 3D vectors produces a third vector that is perpendicular to both u and v. Its magnitude equals |u||v|sin(θ), where θ is the angle between u and v. This makes the cross product sensitive to both the lengths of the vectors and the angle between them — unlike the dot product, which only captures the parallel component.

The direction of u × v is given by the right-hand rule: point your right hand fingers from u toward v (through the smaller angle), and your thumb points in the direction of u × v. Equivalently, u, v, u×v form a right-handed system.

The cross product is anti-commutative: v × u = −(u × v). Swapping the vectors reverses the direction but not the magnitude.

The Determinant Formula

The cross product is computed via the formal determinant:

This is a "formal" determinant — i, j, k are unit vectors, not numbers. Expanding along the first row via cofactors gives the three components.

Worked Example 1: Basic Cross Product

Let u = ⟨1, 0, 0⟩ (the x-axis unit vector) and v = ⟨0, 1, 0⟩ (the y-axis unit vector):

The cross product of the x and y unit vectors is the z unit vector — exactly the right-hand rule for the standard basis.

Worked Example 2: Parallel Vectors

Let u = ⟨2, 4, 6⟩ and v = ⟨1, 2, 3⟩ (note v = u/2, so they are parallel):

Parallel vectors have zero cross product — there is no perpendicular direction to both, and the "parallelogram" they span has zero area (they fall on the same line).

Worked Example 3: Area of a Parallelogram

Find the area of the parallelogram with sides u = ⟨3, −3, 1⟩ and v = ⟨4, 9, 2⟩:

The area of the parallelogram is |u × v| = 5√70. The area of the triangle with these two sides is half that.

Cross Product vs. Dot Product

The dot product u · v is a scalar; the cross product u × v is a vector. The dot product captures how parallel two vectors are; the cross product captures how perpendicular they are.

u · v = 0 means u and v are orthogonal. u × v = 0 means u and v are parallel (or one is zero). Together, they fully characterize the geometric relationship between vectors in 3D.

The identity |u × v|² + (u · v)² = |u|²|v|² (Lagrange identity) connects them: the sum of squared area and squared projection equals the product of squared magnitudes. Use the dot product calculator to compute u · v and verify orthogonality before computing the cross product. For magnitudes like |u × v| = √1750 = 5√70, the simplify radicals calculator reduces the result to simplified radical form.

Applications of the Cross Product

Normal vectors. Given a plane through the origin spanned by u and v, n = u × v is the normal vector. This is used to compute surface normals in 3D graphics for lighting calculations.

Torque. Torque τ = r × F, where r is the position vector and F is the force. The cross product correctly gives the rotational effect perpendicular to both.

Angular momentum. L = r × p, where p = mv is linear momentum. Conservation of angular momentum relies on this vector quantity.

Area computations. |u × v| gives the area of the parallelogram; (1/2)|u × v| gives the area of the triangle. This is used in mesh calculations in computer graphics and finite element analysis.

Volume (scalar triple product). u · (v × w) gives the volume of the parallelepiped spanned by u, v, w. This equals zero if the vectors are coplanar.

The Scalar Triple Product

The scalar triple product u · (v × w) equals the determinant of the 3×3 matrix whose rows are u, v, and w. Its absolute value gives the volume of the parallelepiped spanned by the three vectors.

When the scalar triple product is zero, the three vectors are coplanar — they all lie in the same plane and span no volume. Cyclic permutations leave it unchanged: u·(v×w) = v·(w×u) = w·(u×v). Anti-cyclic permutations negate it.

Example: standard basis vectors e₁ = ⟨1,0,0⟩, e₂ = ⟨0,1,0⟩, e₃ = ⟨0,0,1⟩ give det(I₃) = 1 — the unit cube has volume 1. Use the determinant calculator to compute scalar triple products by entering three vectors as rows.

Cross Product in Physics and Engineering

The cross product models every physical quantity that is perpendicular to two other quantities:

• Torque: τ = r × F, where r is the moment arm and F is force. Magnitude |r||F|sin(θ) measures rotational effectiveness; direction indicates the rotation axis (right-hand rule).
• Angular momentum: L = r × p = r × mv. Conservation of L is a fundamental law of physics. The cross product is what gives spinning tops, gyroscopes, and orbiting planets their stability.
• Magnetic force: F = q(v × B). A moving charge in a magnetic field feels a force perpendicular to its motion — hence circular orbits and cyclotrons.
• Surface normals: Given edges u and v of a triangle in 3D space, n = u × v is the normal vector. All lighting calculations in 3D rendering (OpenGL, WebGL, Vulkan) depend on fast normal computation via the cross product.

In each case, the magnitude |u × v| = |u||v|sin(θ) captures the perpendicular component, while the direction follows the right-hand rule to give a physically meaningful orientation.

Cross Product Identities and the BAC–CAB Rule

The most useful identity for simplifying cross product expressions is the BAC–CAB rule (vector triple product expansion):

This converts a vector triple product — which lies in the plane of b and c — into a linear combination of b and c. The name is a mnemonic for the pattern.

Other important identities:

• Lagrange identity: |u × v|² + (u · v)² = |u|²|v|². Connects cross product and dot product magnitudes.
• Jacobi identity: u × (v × w) + v × (w × u) + w × (u × v) = 0. Makes ℝ³ with cross product a Lie algebra.
• Anticommutativity: u × v = −(v × u). Order matters — reversing swaps the sign.
• Distributivity: u × (v + w) = u × v + u × w.

Use this cross product calculator to verify any of these identities with specific numerical vectors.

Cross Product, Determinants, and Linear Algebra

The cross product formula is a formal determinant expansion — which is why a cross product that comes out as zero tells you the vectors are linearly dependent (they span a 1D subspace instead of 2D). More precisely: u × v = 0 if and only if u and v are parallel (proportional), which is exactly when the 2×3 matrix with rows u and v has rank 1.

This connection to determinants runs deep. The cross product is the unique (up to sign) bilinear, anti-symmetric, dimension-compatible operation in ℝ³. In higher dimensions, the analogous concept is the exterior product u ∧ v, which produces a bivector rather than a vector. The magnitude of the exterior product still gives the area of the parallelogram. The determinant calculator computes the scalar triple product and tests for coplanarity directly. The RREF calculator can verify linear independence of the three vectors by checking their rank. For the broader linear algebra context, see our linear algebra basics guide.

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