Vector Triple Product — Definition, Formula & Examples

The vector triple product is the cross product of one vector with the cross product of two other vectors, written as

\mathbf{a} \times (\mathbf{b} \times \mathbf{c})

. It simplifies using the BAC-CAB identity into a combination of dot products.

For vectors $\mathbf{a}, \mathbf{b}, \mathbf{c} \in \mathbb{R}^3$ , the vector triple product is defined as $\mathbf{a} \times (\mathbf{b} \times \mathbf{c})$ and satisfies the expansion identity $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \mathbf{b}(\mathbf{a} \cdot \mathbf{c}) - \mathbf{c}(\mathbf{a} \cdot \mathbf{b})$ . The result is a vector lying in the plane spanned by $\mathbf{b}$ and $\mathbf{c}$ .

Key Formula

\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \mathbf{b}(\mathbf{a} \cdot \mathbf{c}) - \mathbf{c}(\mathbf{a} \cdot \mathbf{b})

Where:

$\mathbf{a}$ = The first vector (the outer vector in the cross product)
$\mathbf{b}$ = The first vector inside the parenthesized cross product
$\mathbf{c}$ = The second vector inside the parenthesized cross product

How It Works

The key to working with the vector triple product is the BAC-CAB rule. Rather than computing two separate cross products, you expand

\mathbf{a} \times (\mathbf{b} \times \mathbf{c})

into

\mathbf{b}(\mathbf{a} \cdot \mathbf{c}) - \mathbf{c}(\mathbf{a} \cdot \mathbf{b})

, which requires only two dot products and two scalar-vector multiplications. Note that the cross product is not associative:

\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \neq (\mathbf{a} \times \mathbf{b}) \times \mathbf{c}

in general. If the parentheses are on the left instead, you use

(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = \mathbf{b}(\mathbf{a} \cdot \mathbf{c}) - \mathbf{a}(\mathbf{b} \cdot \mathbf{c})

Worked Example

Problem: Compute

\mathbf{a} \times (\mathbf{b} \times \mathbf{c})

where

\mathbf{a} = (1, 2, 0)

\mathbf{b} = (3, 0, 1)

, and

\mathbf{c} = (0, 1, 2)

Compute dot products: Find

\mathbf{a} \cdot \mathbf{c}

and

\mathbf{a} \cdot \mathbf{b}

\mathbf{a} \cdot \mathbf{c} = (1)(0) + (2)(1) + (0)(2) = 2 \qquad \mathbf{a} \cdot \mathbf{b} = (1)(3) + (2)(0) + (0)(1) = 3

Apply BAC-CAB: Substitute into

\mathbf{b}(\mathbf{a} \cdot \mathbf{c}) - \mathbf{c}(\mathbf{a} \cdot \mathbf{b})

2(3, 0, 1) - 3(0, 1, 2) = (6, 0, 2) - (0, 3, 6) = (6, -3, -4)

Answer:

\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (6, -3, -4)

Why It Matters

The vector triple product appears frequently in physics, particularly in electromagnetism (deriving the wave equation from Maxwell's equations) and classical mechanics (rotating reference frames). In linear algebra, it provides a concrete example of why the cross product is not associative, reinforcing careful attention to operation order.

Common Mistakes

Mistake: Assuming the cross product is associative and writing

\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \times \mathbf{b}) \times \mathbf{c}

Correction: These are generally not equal. The BAC-CAB expansion changes depending on which pair is grouped. Always check where the parentheses are before expanding.