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Vector Triple Product

Source lecture(s): PC604 Lec1-2

Intuition

Three vectors multiplied in succession introduce extra terms from expanding determinants.

Formal Definition

The vector triple product identities rearrange products of three vectors.

Mathematical Formulation

\[\vec{A}\times(\vec{B}\times\vec{C})=\vec{B}(\vec{A}\cdot\vec{C})-\vec{C}(\vec{A}\cdot\vec{B})\]

Derivation

Expand component-wise using Levi-Civita symbols \(\epsilon_{ijk}\) and the identity \(\epsilon_{ijk}\epsilon_{klm}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}\).

Worked Example

If \(\vec{A},\vec{C}\) are orthogonal, \(\vec{A}\times(\vec{B}\times\vec{C})\approx -\vec{C}(\vec{A}\cdot\vec{B})\).

Common Mistakes

  • Treating triple product as associative.
  • Dropping the \(-(\vec{A}\cdot\vec{B})\vec{C}\) term.

Quiz

Q1. What is \(\hat{i}\times(\hat{j}\times\hat{k})\)?

Answer

\(\hat{j}(\hat{i}\cdot\hat{k})-\hat{k}(\hat{i}\cdot\hat{j})=0\).