---
title: "Coplanarity of b×c, c×a, a×b"
subject: "Mathematics"
chapter: 5
key: "semester_2/mathematics:coplanarity-b-c-c-b"
source: https://www.bctengineer.com/q/semester_2/mathematics/coplanarity-b-c-c-b
site: BCT Engineering
---

# Coplanarity of b×c, c×a, a×b

Mathematics · Chapter 5: Vector Algebra and Calculus

- Asked in: 2068 Bhadra, 5 marks
- Video: [To prove that three vectors are coplanar (Class 12)](https://www.youtube.com/watch?v=aVQBjpz3o4s) — Conceptual mathematics by Sandeep Wasan
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/coplanarity-b-c-c-b
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-5

## Question

Prove that b×c, c×a and a×b are coplanar or non-coplanar according as a, b, c are coplanar or non-coplanar.

## Short answer

Use [b×c c×a a×b] = [a b c]². If a,b,c are coplanar, [a b c]=0 so the triple product is zero, hence b×c, c×a, a×b are coplanar. If a,b,c are non-coplanar, [a b c]²≠0 so they are non-coplanar.

## Answer

Three vectors are coplanar exactly when their scalar triple product is zero. We use the result

[b×c  c×a  a×b] = [a b c]².

#### Proof of the result
(c × a) × (a × b) = a[(c × a) · b] − b[(c × a) · a] = [a b c] a, using P × (Q × R) = Q(P · R) − R(P · Q) and the fact that c × a is perpendicular to a.

Then [b×c  c×a  a×b] = (b × c) · [(c × a) × (a × b)] = [a b c] (b × c) · a = [a b c]².

#### Coplanar case
If a, b, c are coplanar, [a b c] = 0, so [b×c c×a a×b] = 0² = 0. Hence b × c, c × a, a × b are coplanar.

(Geometrically: when a, b, c lie in one plane, all three cross products are perpendicular to that plane, so they are parallel to each other, hence coplanar.)

#### Non-coplanar case
If a, b, c are non-coplanar, [a b c] ≠ 0, so [b×c c×a a×b] = [a b c]² > 0, which is non-zero. Hence b × c, c × a, a × b are non-coplanar.

**So b×c, c×a, a×b are coplanar if a, b, c are coplanar and non-coplanar if a, b, c are non-coplanar.** (The condition works both ways, since [a b c]² = 0 only when [a b c] = 0.)

## Check yourself

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

1. Three vectors are coplanar exactly when:
   A. their cross product is zero
   B. they are mutually perpendicular
   C. their scalar triple product is zero
   D. their sum is zero
   Answer: C. Zero volume. (Key point: Test)

2. [b×c c×a a×b] equals:
   A. [a b c]
   B. [a b c]²
   C. 2[a b c]
   D. [a b c]³
   Answer: B. Proved via (c×a)×(a×b). (Key point: Key identity)

3. If a, b, c are coplanar, the product is:
   A. 0, so the cross products are coplanar
   B. non-zero
   C. 1
   D. undefined
   Answer: A. 0² = 0. (Key point: Coplanar case)

4. If non-coplanar, [abc]² is:
   A. zero
   B. negative
   C. can be zero
   D. non-zero, so the cross products are non-coplanar
   Answer: D. [abc] ≠ 0 implies [abc]² > 0. (Key point: Non-coplanar)

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- [Coplanar lines and plane containing them](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Acoplanar-lines-plane-containing-them) — Mathematics, chapter 3, asked 2×

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