---
title: "Reciprocal system: sum of cross products"
subject: "Mathematics"
chapter: 5
key: "semester_2/mathematics:reciprocal-system-sum-cross-products"
source: https://www.bctengineer.com/q/semester_2/mathematics/reciprocal-system-sum-cross-products
site: BCT Engineering
---

# Reciprocal system: sum of cross products

Mathematics · Chapter 5: Vector Algebra and Calculus

- Asked in: 2072 Magh, 5 marks
- Video: [Vectors  - ( Reciprocal system of vectors ; Solving problems ) - 84.](https://www.youtube.com/watch?v=MHjF5GkihnU) — Ameen Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/reciprocal-system-sum-cross-products
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-5

## Question

If a, b, c and a', b', c' are the reciprocal system of vectors then prove that a'×b' + b'×c' + c'×a' = (a+b+c)/[a b c], [a b c] ≠ 0.

## Short answer

Using reciprocal vectors a'=(b×c)/[abc] etc., compute a'×b' = c/[abc], b'×c' = a/[abc], c'×a' = b/[abc], and sum to obtain (a+b+c)/[abc].

## Answer

Let [a b c] = V ≠ 0. The reciprocal system is

a′ = (b × c)/V, b′ = (c × a)/V, c′ = (a × b)/V.

#### Step 1: a′ × b′
a′ × b′ = (b × c) × (c × a) / V².

Using P × (Q × R) = Q(P · R) − R(P · Q) with P = b × c, Q = c, R = a:

(b × c) × (c × a) = c[(b × c) · a] − a[(b × c) · c] = V c − 0 = V c.

So a′ × b′ = V c / V² = c/V.

#### Step 2: By cyclic change (a → b → c → a)
```text
b′ × c′ = (c × a) × (a × b) / V² = V a / V² = a/V
c′ × a′ = (a × b) × (b × c) / V² = V b / V² = b/V
```

#### Step 3: Add
a′ × b′ + b′ × c′ + c′ × a′ = (c + a + b)/V.

**Hence a′×b′ + b′×c′ + c′×a′ = (a + b + c)/[a b c].**

## Check yourself

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

1. With V = [a b c], a' × b' equals:
   A. (b × c)·(c × a)/V
   B. (a × b)/V
   C. (b × c)/V²
   D. (b × c) × (c × a)/V²
   Answer: D. Substitute a' and b'. (Key point: Reciprocal definitions)

2. (b × c) × (c × a) equals:
   A. V a
   B. V²c
   C. V c
   D. 0
   Answer: C. From the triple product expansion. (Key point: Key result)

3. So a' × b' equals:
   A. c/V²
   B. c/V
   C. V c
   D. a/V
   Answer: B. V c/V². (Key point: Value)

4. Adding the three cyclic terms gives:
   A. (a + b + c)/V
   B. (a + b + c)/V²
   C. V(a + b + c)
   D. 0
   Answer: A. c/V + a/V + b/V. (Key point: Sum)

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