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Cross Product Calculator

Find the cross product of two 3D vectors with the determinant steps, plus its magnitude, the dot product, the angle between the vectors and the parallelogram area.

Results

ItemValue

Steps

StepWorking
Enter 0 for a missing part. For 2D vectors, set z to 0, and the cross product points along the z-axis. Results are rounded to 6 decimal places.

Quick Answer

The cross product of a = ⟨a₁, a₂, a₃⟩ and b = ⟨b₁, b₂, b₃⟩ is ⟨a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁⟩. For a = ⟨2, 3, 4⟩ and b = ⟨5, 6, 7⟩, a × b = ⟨-3, 6, -3⟩, with magnitude √54 ≈ 7.348469.

Last updated: 2 November 2026 ·How we calculate this: the determinant formula for the cross product

Table of Contents

How This Cross Product Calculator Works

Enter the x, y and z parts of two vectors, a and b. The calculator expands the 3 × 3 determinant with i, j and k on the top row to find a × b, and shows each part. It also gives the magnitude, which equals the area of the parallelogram the vectors make, the dot product, the angle between the vectors and a unit vector perpendicular to both.

Quick explanation: the cross product is a new vector at right angles to both a and b.

  • Direction: follows the right-hand rule.
  • Length: |a||b|sin θ, the parallelogram area.
  • Order matters: b × a = −(a × b).

How to Use the Cross Product Calculator

  1. Enter the x, y and z parts of vector a.
  2. Enter the x, y and z parts of vector b.
  3. Use 0 for any missing part.
  4. Press Calculate.
  5. Read a × b, its magnitude, the angle and the steps.

Core Formulas

  • Cross product: a × b = ⟨a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁⟩
  • Magnitude: |a × b| = |a| |b| sin θ
  • Dot product: a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a| |b| cos θ
  • Triangle area: ½ |a × b|
  • Unit normal: (a × b) ÷ |a × b|

Key Terms

  • Vector: a quantity with size and direction, written ⟨x, y, z⟩.
  • Cross product: a vector perpendicular to two vectors, with length equal to their parallelogram area.
  • Dot product: a number equal to |a||b|cos θ.
  • Magnitude: the length of a vector.
  • Right-hand rule: point fingers along a, curl toward b, and your thumb shows a × b.

Worked Example

Find a × b for a = ⟨1, 0, 0⟩ and b = ⟨0, 1, 0⟩.

  • i part: 0 × 0 − 0 × 1 = 0
  • j part: 0 × 0 − 1 × 0 = 0
  • k part: 1 × 1 − 0 × 0 = 1, so a × b = ⟨0, 0, 1⟩, which is the z-axis

Real-Life Examples

  • Physics: torque is τ = r × F, and magnetic force is F = qv × B.
  • Area: the triangle with sides a = ⟨2, 3, 4⟩ and b = ⟨5, 6, 7⟩ has area 3.674235.
  • 3D graphics: the normal to a surface comes from the cross product of two edges.
  • Plane equation: the normal of a plane through three points is the cross product of two edge vectors.

Cross Product vs Dot Product

  • Cross product: gives a vector, largest when the vectors are perpendicular.
  • Dot product: gives a number, largest when the vectors point the same way.
  • Zero results: cross is zero for parallel vectors, dot is zero for perpendicular ones.

Cross Products of Unit Vectors

Read across: row × column. For example, i × j = k and j × i = −k.

×ijk
i0k−j
j−k0i
kj−i0

Key Properties

  • Anti-commutative: a × b = −(b × a).
  • Parallel vectors: the cross product is the zero vector.
  • Perpendicular: (a × b) · a = 0 and (a × b) · b = 0.
  • Only in 3D: the usual cross product is defined for three-dimensional vectors.

Common Mistakes to Avoid

  • Swapping the order. b × a points the opposite way to a × b.
  • Forgetting the minus on the j part. With cofactor expansion, the j term is subtracted.
  • Mixing up cross and dot. One gives a vector, the other a number.
  • Using 2D vectors directly. Add z = 0 first.
  • Taking the magnitude as the area of a triangle. The triangle is half the parallelogram.

Frequently Asked Questions

Use ⟨a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁⟩, or expand the determinant with i, j and k.

A vector at right angles to both vectors, with length equal to the parallelogram area they span.

No. a × b = −(b × a).

The zero vector, since sin 0° = 0.

Set z = 0. The result points along the z-axis, with z-part a₁b₂ − a₂b₁.

Use cos θ = (a · b) ÷ (|a| |b|). The calculator shows it.

|a| |b| sin θ.

Half the magnitude of the cross product of two side vectors.

k. Also j × k = i and k × i = j.

With your right hand, point along a and curl your fingers toward b. Your thumb points along a × b.

In physics for torque and magnetic force, and in 3D graphics for surface normals.

Yes. They are two names for the same operation.

References & Sources

SourceUsed ForLink
Wikipedia, Cross product Definition, determinant form and properties. https://en.wikipedia.org/wiki/Cross_product
OpenStax, Calculus Volume 3, 2.4 The Cross Product Free textbook section with worked examples. https://openstax.org/books/calculus-volume-3/pages/2-4-the-cross-product