---
title: "Reciprocal system: a·a'+b·b'+c·c'=3"
subject: "Mathematics"
chapter: 5
key: "semester_2/mathematics:reciprocal-system-b-b-c"
source: https://www.bctengineer.com/q/semester_2/mathematics/reciprocal-system-b-b-c
site: BCT Engineering
---

# Reciprocal system: a·a'+b·b'+c·c'=3

Mathematics · Chapter 5: Vector Algebra and Calculus

- Asked in: 2071 Bhadra, 5 marks
- Video: [✍️ Reciprocal System of Vectors ✍️| Vector Calculus BA/BSc Maths | Engineering Maths | New Era Maths](https://www.youtube.com/watch?v=db4SQGOc4Cg) — New Era Maths Classes
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/reciprocal-system-b-b-c
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-5

## Question

If a, b, c and a', b', c' are reciprocal system of vectors, then show that a·a' + b·b' + c·c' = 3.

## Short answer

By definition, a·a' = a·(b×c)/[abc] = 1, and similarly b·b' = 1, c·c' = 1, so the sum is 3.

## Answer

Let V = [a b c] ≠ 0. The reciprocal system is a′ = (b × c)/V, b′ = (c × a)/V, c′ = (a × b)/V.

#### Each dot product
```text
a · a′ = a · (b × c) / V = [a b c]/V = 1
b · b′ = b · (c × a) / V = [b c a]/V = [a b c]/V = 1
c · c′ = c · (a × b) / V = [c a b]/V = [a b c]/V = 1
```
Here [b c a] = [c a b] = [a b c] by cyclic permutation of the triple product.

#### Add
a · a′ + b · b′ + c · c′ = 1 + 1 + 1.

**Hence a · a′ + b · b′ + c · c′ = 3.**

## Check yourself

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

1. The reciprocal vector a' is:
   A. (b × c)/V, V = [a b c]
   B. (a × b)/V
   C. (b · c)/V
   D. b × c
   Answer: A. Standard definition. (Key point: Definition)

2. a · a' equals:
   A. 0
   B. V
   C. 1/V
   D. [a b c]/V = 1
   Answer: D. a · (b × c) = V. (Key point: Each dot)

3. Why does b · b' also equal 1?
   A. b = a
   B. [b c a] = −[a b c]
   C. [b c a] = [a b c] by cyclic permutation
   D. b' = a'
   Answer: C. Triple product is cyclic. (Key point: Cyclic)

4. The sum is:
   A. 1
   B. 3
   C. 0
   D. V
   Answer: B. 1 + 1 + 1. (Key point: Sum)

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