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Vector Triple Product

A×(B×C) = B(A·C) − C(A·B) (BAC-CAB rule). Result lies in plane of B and C. Scalar triple A·(B×C) = signed volume of parallelepiped. Cyclic: A·(B×C) = B·(C×A) = C·(A×B).

Concept Fundamentals
A×(B×C) = B(A·C) − C(A·B)
BAC-CAB
A·(B×C) = volume
Scalar
A·(B×C) = B·(C×A)
Cyclic
Lies in plane of B, C
Plane

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BAC-CAB: A×(B×C) = B(A·C) − C(A·B). Scalar triple = signed volume of parallelepiped. Coplanar vectors: scalar triple = 0.

Key quantities
A×(B×C) = B(A·C) − C(A·B)
BAC-CAB
Key relation
A·(B×C) = volume
Scalar
Key relation
A·(B×C) = B·(C×A)
Cyclic
Key relation
Lies in plane of B, C
Plane
Key relation

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Why: Vector triple product appears in angular momentum (L = r×p), electromagnetism, and rigid body dynamics. Scalar triple gives volume; vector triple gives a vector in the B-C plane.

How: Scalar: compute B×C first, then A·(B×C). Vector: use BAC-CAB: A×(B×C) = B(A·C) − C(A·B). Scalar triple = 0 when vectors are coplanar.

BAC-CAB: A×(B×C) = B(A·C) − C(A·B).Scalar triple = signed volume of parallelepiped.
Sources:Khan Academy — Vector Triple ProductWolfram MathWorld — Vector Triple Product

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Compute Triple ProductsEnter three 3D vectors

Three 3D Vectors

Vector A

Vector B

Vector C

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🧮 Fascinating Math Facts

×

A×(B×C) = B(A·C) − C(A·B) (BAC-CAB).

— Vector Triple

V

A·(B×C) = signed volume.

— Scalar Triple

Key Takeaways

  • • Scalar triple product A·(B×C) = volume of parallelepiped formed by A, B, C. Zero if coplanar.
  • • Vector triple product A×(B×C) = B(A·C) - C(A·B) — lies in the plane of B and C.
  • • A×(B×C) ≠ (A×B)×C in general. Order matters.
  • • Jacobi identity: A×(B×C) + B×(C×A) + C×(A×B) = 0.
  • • Scalar triple product is cyclic: A·(B×C) = B·(C×A) = C·(A×B).

Did You Know?

Volume Interpretation

|A·(B×C)| is the volume of the parallelepiped with edges A, B, C. The sign indicates orientation.

BAC-CAB Rule

The identity A×(B×C) = B(A·C) - C(A·B) is often called the "BAC-CAB" rule for the order of vectors.

Coplanarity

If A·(B×C)=0, the three vectors are coplanar (lie in the same plane).

Electromagnetism

The vector triple product appears in the Lorentz force and in expressions like v×(B×j) in MHD.

Rotation

In rigid body dynamics, ω×(r×v) appears when computing angular momentum and torques.

Cross Product Non-Associativity

Cross product is not associative: (A×B)×C ≠ A×(B×C). The vector triple product identity helps simplify.

Understanding Triple Products

The scalar triple product A·(B×C) gives the signed volume of the parallelepiped. The vector triple product A×(B×C) expands as:

A⃗×(B⃗×C⃗)=B⃗(A⃗⋅C⃗)−C⃗(A⃗⋅B⃗)\vec{A} \times (\vec{B} \times \vec{C}) = \vec{B}(\vec{A} \cdot \vec{C}) - \vec{C}(\vec{A} \cdot \vec{B})

Expert Tips

Volume Zero

If A·(B×C)=0, the vectors are coplanar. No parallelepiped volume.

BAC-CAB

Memorize: A×(B×C) = B(A·C) - C(A·B). The result is in the plane of B and C.

Cyclic Property

A·(B×C) = B·(C×A) = C·(A×B). Swapping cyclically preserves the value.

Right-Hand Rule

Positive scalar triple means A, B, C form a right-handed system.

FAQ

What is the scalar triple product?

A·(B×C) is the dot product of A with the cross product B×C. It equals the signed volume of the parallelepiped with edges A, B, C.

What is the vector triple product?

A×(B×C) = B(A·C) - C(A·B). It is a vector in the plane of B and C.

When is A·(B×C) = 0?

When A, B, C are coplanar (linearly dependent). The parallelepiped collapses to zero volume.

Is (A×B)×C = A×(B×C)?

No. Cross product is not associative. Use the BAC-CAB rule to expand A×(B×C).

What does positive/negative volume mean?

Positive: right-handed system. Negative: left-handed. |A·(B×C)| is the actual volume.

Why is it called BAC-CAB?

From the identity: A×(B×C) = B(A·C) - C(A·B). The vector names spell BAC and CAB.

Where is this used?

Physics (torque, angular momentum), electromagnetism, fluid dynamics, and 3D geometry.

How to Use

  1. Enter components of three 3D vectors A, B, and C.
  2. Click a sample example or Calculate.
  3. Review scalar triple product (volume), vector triple product, and steps.
  4. Copy results if needed.

Disclaimer: All three vectors must be 3D. Results use standard floating-point arithmetic.

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