---
title: "Scalar triple product identity"
subject: "Mathematics"
chapter: 5
key: "semester_2/mathematics:scalar-triple-product-identity"
source: https://www.bctengineer.com/q/semester_2/mathematics/scalar-triple-product-identity
site: BCT Engineering
---

# Scalar triple product identity

Mathematics · Chapter 5: Vector Algebra and Calculus

- Asked in: 2075 Baisakh, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/scalar-triple-product-identity
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-5

## Question

Prove that [a×b c×d e×f] = [a b d][c e f] − [a b c][d e f].

## Short answer

Use (c×d)×(e×f) = [c e f] d − [d e f] c. Dot with (a×b) gives [a×b c×d e×f] = [a b d][c e f] − [a b c][d e f].

## Answer

We use [u v w] = (u × v) · w and (P × Q) × R = Q(P · R) − P(Q · R).

#### Step 1: Expand (c × d) × (e × f)
Put Y = e × f. Then

(c × d) × Y = d(c · Y) − c(d · Y), with c · Y = c · (e × f) = [c e f] and d · Y = [d e f].

So (c × d) × (e × f) = [c e f] d − [d e f] c.

#### Step 2: Take the triple product
[a×b  c×d  e×f] = (a × b) · [(c × d) × (e × f)]

```text
= (a × b) · ([c e f] d − [d e f] c)
= [c e f] (a × b) · d − [d e f] (a × b) · c
= [c e f] [a b d] − [d e f] [a b c]
```

**Hence [a×b c×d e×f] = [a b d][c e f] − [a b c][d e f].**

(Check with a = i, b = j, c = i, d = k, e = j, f = k: a×b = k, c×d = −j, e×f = i, so the left side is k · ((−j) × i) = k · k = 1. The right side is [i j k][i j k] − [i j i][k j k] = 1 − 0 = 1 ✓.)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. (c × d) × (e × f) equals:
   A. [c d e] f − [c d f] e
   B. [c e f] c − [d e f] d
   C. [c e f] d − [d e f] c
   D. [c d f] e
   Answer: C. Put Y = e × f and apply the triple product rule. (Key point: Expansion)

2. (a × b) · d equals:
   A. [a d b]
   B. [a b d]
   C. a · d
   D. 0
   Answer: B. By definition of the triple product. (Key point: Dot product)

3. Combining the terms gives:
   A. [a b d][c e f] − [a b c][d e f]
   B. [a b c][d e f] − [a b d][c e f]
   C. [a b d][c e f] + [a b c][d e f]
   D. [a b c][c e f]
   Answer: A. Dotting the expansion with a × b. (Key point: Final result)

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