This cross product calculator computes a × b for two 3D vectors — the vector perpendicular to both, with its magnitude (the parallelogram area) and the angle between the original vectors.

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The Perpendicular, With an Area Attached

The cross product is the vector operation that answers “what’s perpendicular to both of these, and how much do they spread?” For a = (1, 2, 3) and b = (4, 5, 6), the determinant formula gives a × b = (2·6 − 3·5, 3·4 − 1·6, 1·5 − 2·4) = (−3, 6, −3), magnitude √54 ≈ 7.348, and the vectors meet at 12.93°. Three facts are welded into that output: the direction is orthogonal to both inputs (check: (−3)(1) + 6·2 + (−3)(3) = 0 ✓); the magnitude equals |a||b|sinθ — the area of the parallelogram spanned, which is why torque (r × F), angular momentum (r × p) and the magnetic force (q v × B) are cross products, each needing a perpendicular direction whose size scales with the spread between the factors; and the operation is anti-commutative, a × b = −(b × a), because “perpendicular” has two directions and the right-hand rule must pick one consistently. It exists only in 3 dimensions (and, by a quirk, 7) — in 2D only the z-component survives as a scalar, the signed area, which is exactly the 2D determinant and the engine of the shoelace area formula.

The mechanics deserve care because sign errors breed here: the middle component carries the minus — a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁), with the y-term’s subtraction in the “reversed” order, equivalently the formal determinant with the middle cofactor negated. The calculator’s verification is built in: dot the answer with either input and you must get zero; if you don’t, a sign slipped. Applications map the territory of 3D work: surface normals in graphics (two edge vectors cross into the face normal that lighting depends on), torque and angular momentum in mechanics, the Lorentz force in electromagnetism, and the scalar triple product a·(b × c) — the parallelepiped’s volume — which the magnitude here is two-thirds of. Where the dot product measures alignment (a scalar: how much together), the cross measures anti-alignment (a vector: how much apart, and around which axis) — the pair of them exhaust what two vectors can say about each other.

Worked Examples (Verified)

  • Defaults: (1,2,3) × (4,5,6) = (−3, 6, −3), |·| = 7.348, angle 12.93°.
  • Unit axes: î × ĵ = k̂ — the right-hand rule’s founding example; ĵ × î = −k̂.
  • Parallel vectors: (1,2,3) × (2,4,6) = (0, 0, 0) — no spread, no product.
  • Perpendicular pair: (2,0,0) × (0,3,0) = (0, 0, 6) — magnitude 2 × 3 = the rectangle’s area.

Cross Product Discipline

  • The middle component’s sign is the classic error: use the determinant layout and negate the y-cofactor, or verify with the dot-product-equals-zero check every time.
  • a × b ≠ b × a: swapping inputs flips every sign. Physics formulas (torque = r × F, not F × r) encode a convention — respect the order given.
  • Cross product is 3D-only as a vector; in 2D problems use the scalar z-component (a₁b₂ − a₂b₁) for areas and orientation tests.
  • Distributivity holds, associativity does NOT: (a × b) × c ≠ a × (b × c) generally — parentheses are load-bearing.
  • Zero cross product means parallel, collinear, or a zero input — diagnose which before interpreting.

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Frequently Asked Questions

How do you calculate the cross product?

Componentwise, as a determinant: a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). For (1,2,3) × (4,5,6): (12−15, 12−6, 5−8) = (−3, 6, −3). The mnemonic is the 3×3 determinant with î, ĵ, k̂ in the top row — remembering that the middle cofactor takes a minus sign. This calculator returns the vector, its magnitude and the angle between the inputs, and the orthogonality check (result · a = result · b = 0) verifies any hand computation.

What does the cross product represent geometrically?

A vector perpendicular to both inputs whose length is the area of the parallelogram they span: |a × b| = |a||b|sinθ. Direction by the right-hand rule — point fingers along a, curl toward b, thumb gives a × b. Halve the magnitude and you have the triangle’s area; dot it with a third vector and you have a parallelepiped’s volume (scalar triple product). Torque, angular momentum and magnetic force are cross products because nature keeps needing exactly this: a perpendicular effect sized by the spread of two causes.

Why is the cross product only defined in 3D?

Because “the direction perpendicular to a plane” is unique (up to sign) only in 3D: two independent vectors span a plane, and in 3D exactly one axis is orthogonal to it. In 2D the would-be result points out of the page — captured as a scalar (the determinant). In 4D+ a plane has infinitely many perpendicular directions, so no unique vector exists (though the wedge product generalizes the area part). A 7-dimensional exception exists via the octonions — a beloved curiosity with no engineering applications, which tells you how special 3D is.

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

Everything dual: dot returns a SCALAR, a·b = |a||b|cosθ, measuring alignment (maximal when parallel, zero when perpendicular, commutative). Cross returns a VECTOR, |a × b| = |a||b|sinθ in the perpendicular direction, measuring spread (zero when parallel, maximal when perpendicular, anti-commutative). Together they’re exhaustive: cos from the dot, sin from the cross’s magnitude, and Lagrange’s identity welds them: |a × b|² + (a·b)² = |a|²|b|² — use it to cross-check computations, as the angle this calculator reports does via both routes.

What does a zero cross product mean?

The vectors fail to span any area: they are parallel or antiparallel (collinear), or at least one is the zero vector. Far from an error, it’s a standard test — collinearity of points, parallelism of lines and planes, and linear dependence all reduce to cross products vanishing. In physics it carries meaning too: zero torque when force acts along the lever, zero magnetic force on a charge moving parallel to the field. A zero here is a theorem about your inputs, delivered free.

How is the cross product used for torque?

τ = r × F: the lever arm vector (pivot to application point) crossed with the force. Magnitude |r||F|sinθ captures the bench wisdom — push perpendicular at the end of the wrench (θ = 90°, r max) for full torque; push along the wrench (θ = 0) and nothing turns, exactly the sinθ = 0 of the formula. Direction is the rotation axis by the right-hand rule. This calculator’s machinery is literally the torque computation with generic inputs — substitute r and F and read the axis and magnitude.

What is the scalar triple product?

a · (b × c): the dot of one vector with the cross of the other two, equal to the (signed) volume of the parallelepiped they span, and to the 3×3 determinant of their components. It vanishes exactly when the three vectors are coplanar — the standard flatness/dependence test. From this page’s outputs it’s one extra step: compute b × c here, then dot with a by hand or symmetry. Volume of a tetrahedron is |triple|/6 — three of the most useful formulas in 3D geometry stacked on one operation.

Is this cross product calculator free?

Yes — free, instant, no sign-up. Enter the six components for a × b, its magnitude and the angle between the vectors.