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Vector Cross Product

A×B is a vector perpendicular to both A and B. |A×B| = |A||B|sin θ = area of parallelogram. Right-hand rule gives direction. Component form: (aᵧb_z−a_zbᵧ, a_zbₓ−aₓb_z, aₓbᵧ−aᵧbₓ).

Concept Fundamentals
Perpendicular to A and B
A×B
|A×B| = |A||B|sin θ
Magnitude
Parallelogram area
Area
Direction rule
Right-hand

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A×B = −B×A (anticommutative). |A×B| = area of parallelogram. Parallel vectors: A×B = 0.

Key quantities
Perpendicular to A and B
A×B
Key relation
|A×B| = |A||B|sin θ
Magnitude
Key relation
Parallelogram area
Area
Key relation
Direction rule
Right-hand
Key relation

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Why: Cross product gives torque, angular momentum, surface normals in 3D graphics, and area of parallelograms. Right-hand rule determines direction.

How: A×B = (aᵧb_z−a_zbᵧ, a_zbₓ−aₓb_z, aₓbᵧ−aᵧbₓ). Magnitude = |A||B|sin θ. A×B is perpendicular to both; A·(A×B) = B·(A×B) = 0.

A×B = −B×A (anticommutative).|A×B| = area of parallelogram.
Sources:Khan Academy — Cross ProductWolfram MathWorld — Cross Product

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Compute Cross ProductEnter two 3D vectors

Enter Two 3D Vectors

Vector A

Vector B

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

×

A×B perpendicular to A and B.

— Cross Product

▱

|A×B| = area of parallelogram.

— Geometry

Key Takeaways

  • • The cross product A × B is a vector perpendicular to both A and B.
  • • vecAtimesvecB=(aybz−azby,azbx−axbz,axby−aybx)\\vec{A} \\times \\vec{B} = (a_y b_z - a_z b_y,\\ a_z b_x - a_x b_z,\\ a_x b_y - a_y b_x).
  • • ∣vecAtimesvecB∣=∣vecA∣∣vecB∣sintheta|\\vec{A} \\times \\vec{B}| = |\\vec{A}||\\vec{B}|\\sin\\theta = area of the parallelogram with sides A and B.
  • • Anticommutative: A × B = -(B × A).
  • • For parallel vectors, A × B = 0.

Did You Know?

Physics

Torque τ = r × F. Angular momentum L = r × p. The cross product is fundamental in rotational dynamics.

Right-Hand Rule

Point fingers in direction of A, curl toward B. Thumb points in direction of A × B.

Computer Graphics

Cross products compute surface normals for lighting. Used to determine triangle orientation.

3D Only

The cross product is defined only in 3D. In 2D, the result would need a 3rd dimension.

Area of Triangle

Area of triangle with sides A and B = ½|A × B|.

Standard Basis

i × j = k, j × k = i, k × i = j. Cyclic order gives positive result.

Understanding the Cross Product

The cross product of two 3D vectors is a vector perpendicular to both:

A⃗×B⃗=(aybz−azbyazbx−axbzaxby−aybx)\vec{A} \times \vec{B} = \begin{pmatrix} a_y b_z - a_z b_y \\ a_z b_x - a_x b_z \\ a_x b_y - a_y b_x \end{pmatrix}

Its magnitude equals the area of the parallelogram with sides A and B: |A×B| = |A||B|sin θ.

Expert Tips

Right-Hand Rule

Use the right-hand rule to determine direction: A × B points where your thumb points when fingers curl from A to B.

Parallel Check

If A × B = 0 and neither is zero, A and B are parallel (or antiparallel).

Perpendicularity

Verify (A×B)·A = 0 and (A×B)·B = 0. The cross product is always perpendicular to both.

Area of Triangle

For a triangle with two sides as vectors A and B, area = ½|A × B|.

Frequently Asked Questions

What is the cross product?

A vector perpendicular to both A and B. A × B = (a_y b_z - a_z b_y, a_z b_x - a_x b_z, a_x b_y - a_y b_x).

Why is it only defined in 3D?

In 3D, the space of vectors perpendicular to a plane is 1D. In 2D, the result would need a 3rd dimension.

How do I find the direction?

Use the right-hand rule: fingers from A to B, thumb points in direction of A × B.

When is A × B = 0?

When A and B are parallel (or one is zero). Then sin θ = 0.

What is the magnitude?

|A × B| = |A||B|sin θ = area of the parallelogram with sides A and B.

Is A × B = B × A?

No. A × B = -(B × A). The cross product is anticommutative.

How do I verify perpendicularity?

Check (A×B)·A = 0 and (A×B)·B = 0. Both should be zero (within floating-point tolerance).

How to Use This Calculator

  1. Enter components of two 3D vectors A and B.
  2. Click "Calculate" to get the cross product, its magnitude, area of parallelogram, and perpendicularity checks.
  3. Review the visualization showing A, B, and A×B.
  4. Copy results for homework or reports.

Disclaimer: This calculator uses standard floating-point arithmetic. The cross product is defined only for 3D vectors. Results are suitable for educational and professional use.

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