Vector Triple Product: Formula, Proof and Examples

The vector triple product a × (b × c) is the cross product of a with the cross product of b and c. Unlike the scalar triple product, its result is a vector — specifically, a vector that lies in the plane of b and c.

It matters because nested cross products appear in physics, for example in rotation and electromagnetism, and the BAC–CAB formula collapses them into a simple combination of b and c, avoiding a messy double computation.

Key Concepts

• Definition: a × (b × c), where the brackets matter — cross products are not associative, so bracket position changes the answer.
• The result is always coplanar with b and c: it can be written using only b and c, with no part along any other direction.
• BAC–CAB rule: a × (b × c) = (a · c) b − (a · b) c. Read it as "back minus cab" to memorize it.
• The mirrored form: (a × b) × c = (a · c) b − (b · c) a. Note how the bracket position changes the factored terms.
• a × (b × c) = 0 when b and c are parallel (so b × c = 0), or when a lies in the plane of b and c.
• The Lagrange identity follows from it: |a × b|² = |a|²|b|² − (a · b)².

Formulas

• a × (b × c) = (a · c) b − (a · b) c — the BAC–CAB rule.
• (a × b) × c = (a · c) b − (b · c) a.
• Proof idea: both sides are linear in each of a, b, c, so it is enough to verify the formula on the basis vectors i, j, k. Direct expansion confirms it there, hence it holds for all vectors.
• Consequence: since the right side involves only b and c, a × (b × c) is perpendicular to b × c.

Worked Example

Compute a × (b × c) for a = (1, 0, 2), b = (3, 1, 0), c = (0, 1, 4).

1. Dot products: a · c = 1×0 + 0×1 + 2×4 = 8; a · b = 1×3 + 0×1 + 2×0 = 3.
2. BAC–CAB: a × (b × c) = 8b − 3c = 8(3, 1, 0) − 3(0, 1, 4) = (24, 8, 0) − (0, 3, 12) = (24, 5, −12).
3. Verify directly: b × c = (1×4 − 0×1, 0×0 − 3×4, 3×1 − 1×0) = (4, −12, 3).
4. a × (4, −12, 3): x = 0×3 − 2×(−12) = 24; y = 2×4 − 1×3 = 5; z = 1×(−12) − 0×4 = −12, giving (24, 5, −12). Matches the formula.

Common Misconceptions

• a × (b × c) = (a × b) × c. False — the cross product is not associative, and the two bracketings give different vectors in general.
• The result is parallel to a. Wrong: it lies in the plane of b and c, expressed as (a · c) b − (a · b) c.
• Brackets can be dropped. Dropping them makes the expression ambiguous; always write which cross is done first.

Real-World Applications

• Physics: the BAC–CAB rule simplifies nested cross products in rotational dynamics and electromagnetism.
• Vector calculus identities, such as the double-curl formula, are proved with it.
• The Lagrange identity |a × b|² = |a|²|b|² − (a · b)², used in proofs and error bounds.
• Computer graphics: decomposing vectors relative to planes.

The vector triple product a × (b × c) always lands in the plane of b and c and is given by the BAC–CAB rule (a · c) b − (a · b) c. Memorize the mnemonic, respect the brackets, and nested cross products become routine.