---
title: "Reciprocal system of −i+j+k, i−j+k, i+j−k"
subject: "Mathematics"
chapter: 5
key: "semester_2/mathematics:reciprocal-system-i-j-k"
source: https://www.bctengineer.com/q/semester_2/mathematics/reciprocal-system-i-j-k
site: BCT Engineering
---

# Reciprocal system of −i+j+k, i−j+k, i+j−k

Mathematics · Chapter 5: Vector Algebra and Calculus

- Asked in: 2068 Bhadra, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/reciprocal-system-i-j-k
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-5

## Question

Find a set of vectors reciprocal to the following vectors: −i+j+k, i−j+k, i+j−k.

## Short answer

Triple product [a b c] = 4. Cross products: b×c = (0,2,2), c×a = (2,0,2), a×b = (2,2,0). Dividing by 4 gives reciprocal vectors a′ = (j+k)/2, b′ = (i+k)/2, c′ = (i+j)/2.

## Answer

For a, b, c with [a b c] ≠ 0: a′ = (b × c)/[a b c], b′ = (c × a)/[a b c], c′ = (a × b)/[a b c].

a = (−1, 1, 1), b = (1, −1, 1), c = (1, 1, −1).

#### Cross products
```text
b × c = ((−1)(−1) − (1)(1), (1)(1) − (1)(−1), (1)(1) − (−1)(1))
      = (1 − 1, 1 + 1, 1 + 1) = (0, 2, 2)
c × a = ((1)(1) − (−1)(1), (−1)(−1) − (1)(1), (1)(1) − (1)(−1))
      = (1 + 1, 1 − 1, 1 + 1) = (2, 0, 2)
a × b = ((1)(1) − (1)(−1), (1)(1) − (−1)(1), (−1)(−1) − (1)(1))
      = (1 + 1, 1 + 1, 1 − 1) = (2, 2, 0)
```

#### Triple product
[a b c] = a · (b × c) = (−1)(0) + (1)(2) + (1)(2) = 4.

#### Reciprocal vectors
```text
a′ = (0, 2, 2)/4 = (j + k)/2
b′ = (2, 0, 2)/4 = (i + k)/2
c′ = (2, 2, 0)/4 = (i + j)/2
```

Check: a · a′ = (0 + 1 + 1)/2 = 1 ✓; b · b′ = (1 + 0 + 1)/2 = 1 ✓; c · c′ = (1 + 1 + 0)/2 = 1 ✓; a · b′ = (−1 + 0 + 1)/2 = 0 ✓.

**Answer: a′ = (j + k)/2, b′ = (k + i)/2, c′ = (i + j)/2.**

## Check yourself

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

1. [a b c] for a = (−1,1,1), b = (1,−1,1), c = (1,1,−1) is:
   A. −4
   B. 4
   C. 2
   D. 0
   Answer: B. a · (0,2,2) = 4. (Key point: Triple product)

2. b × c equals:
   A. (0, 2, 2)
   B. (2, 0, 2)
   C. (0, −2, 2)
   D. (2, 2, 0)
   Answer: A. Determinant. (Key point: b × c)

3. a' equals:
   A. (i + k)/2
   B. (j + k)/4
   C. (j + k)
   D. (j + k)/2
   Answer: D. (0,2,2)/4. (Key point: a')

## Close variants set in other exams

- [Reciprocal system of 2i+3j−k, i−j−2k, −i+2j+2k](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Areciprocal-system-2i-3j-k-2) — Mathematics, chapter 5, asked 2×

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