---
title: "Reciprocal system of 2i+3j−k, i−j−2k, −i+2j+2k"
subject: "Mathematics"
chapter: 5
key: "semester_2/mathematics:reciprocal-system-2i-3j-k-2"
source: https://www.bctengineer.com/q/semester_2/mathematics/reciprocal-system-2i-3j-k-2
site: BCT Engineering
---

# Reciprocal system of 2i+3j−k, i−j−2k, −i+2j+2k

Mathematics · Chapter 5: Vector Algebra and Calculus

- Asked in: 2070 Bhadra, 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-2i-3j-k-2
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-5

## Question

Find a set of vectors reciprocal to the following vectors 2i+3j−k, i−j−2k, −i+2j+2k.

## Short answer

Compute b×c = (2,0,1), c×a = (-8,3,-7), a×b = (-7,3,-5); [a b c] = 3. Reciprocal vectors: a' = (2i + k)/3, b' = (-8i + 3j - 7k)/3, c' = (-7i + 3j - 5k)/3.

## Answer

For a, b, c with [a b c] ≠ 0, the reciprocal set is a′ = (b × c)/[a b c], b′ = (c × a)/[a b c], c′ = (a × b)/[a b c].

Here a = (2, 3, −1), b = (1, −1, −2), c = (−1, 2, 2).

#### Cross products
```text
b × c = ((−1)(2) − (−2)(2), (−2)(−1) − (1)(2), (1)(2) − (−1)(−1))
      = (−2 + 4, 2 − 2, 2 − 1) = (2, 0, 1)
c × a = ((2)(−1) − (2)(3), (2)(2) − (−1)(−1), (−1)(3) − (2)(2))
      = (−2 − 6, 4 − 1, −3 − 4) = (−8, 3, −7)
a × b = ((3)(−2) − (−1)(−1), (−1)(1) − (2)(−2), (2)(−1) − (3)(1))
      = (−6 − 1, −1 + 4, −2 − 3) = (−7, 3, −5)
```

#### Triple product
[a b c] = a · (b × c) = 2(2) + 3(0) + (−1)(1) = 3.

#### Reciprocal vectors
```text
a′ = (2i + k)/3
b′ = (−8i + 3j − 7k)/3
c′ = (−7i + 3j − 5k)/3
```

Check: a · a′ = (4 − 1)/3 = 1 ✓; b · b′ = (−8 − 3 + 14)/3 = 1 ✓; c · c′ = (7 + 6 − 10)/3 = 1 ✓; a · b′ = (−16 + 9 + 7)/3 = 0 ✓.

**Answer: a′ = (2i + k)/3, b′ = (−8i + 3j − 7k)/3, c′ = (−7i + 3j − 5k)/3.**

## Check yourself

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

1. [a b c] for a = (2,3,−1), b = (1,−1,−2), c = (−1,2,2) is:
   A. −3
   B. 3
   C. 1
   D. 6
   Answer: B. a · (2,0,1) = 4 + 0 − 1 = 3. (Key point: Triple product)

2. b × c equals:
   A. (2, 0, 1)
   B. (−2, 0, 1)
   C. (2, 1, 0)
   D. (−8, 3, −7)
   Answer: A. Determinant. (Key point: b × c)

3. a' equals:
   A. (2i + k)
   B. (2i + k)/6
   C. (−8i + 3j − 7k)/3
   D. (2i + k)/3
   Answer: D. (b×c)/3. (Key point: a')

4. A check on b' is:
   A. b · b' = 0
   B. a · b' = 1
   C. b · b' = 1
   D. b × b' = 0
   Answer: C. (−8 − 3 + 14)/3 = 1. (Key point: Check)

## Close variants set in other exams

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

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