---
title: "Divergence of (a×r)/r is zero"
subject: "Mathematics"
chapter: 5
key: "semester_2/mathematics:divergence-r-r-zero"
source: https://www.bctengineer.com/q/semester_2/mathematics/divergence-r-r-zero
site: BCT Engineering
---

# Divergence of (a×r)/r is zero

Mathematics · Chapter 5: Vector Algebra and Calculus

- Asked in: 2072 Magh, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/divergence-r-r-zero
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-5

## Question

If r be the position vector and a is constant vector then prove that ∇·((a×r)/r) = 0.

## Short answer

Apply divergence product rule: ∇(1/r) = -r/r³ and ∇·(a×r)=0; thus ∇·((a×r)/r) = (-r/r³)·(a×r) = 0, since r·(a×r)=0.

## Answer

Let r = xi + yj + zk, r = |r|, a a constant vector. Use ∇ · (φF) = ∇φ · F + φ(∇ · F) with φ = 1/r and F = a × r.

#### ∇φ
∇(1/r) = −(1/r²)∇r = −(1/r²)(r/r) = −r/r³.

#### ∇ · (a × r)
Using ∇ · (a × r) = r · (∇ × a) − a · (∇ × r). Here ∇ × a = 0 (a is constant) and ∇ × r = 0, so ∇ · (a × r) = 0.

(Direct check: a × r = (a₂z − a₃y, a₃x − a₁z, a₁y − a₂x); ∂/∂x of the first is 0, ∂/∂y of the second is 0, ∂/∂z of the third is 0.)

#### Combine
```text
∇ · ((a × r)/r) = (−r/r³) · (a × r) + (1/r)(0)
                = −(1/r³) r · (a × r) + 0
                = −(1/r³) [r a r]  = 0
```
because r · (a × r) is a scalar triple product with a repeated vector r, so it is zero (r is perpendicular to a × r).

**Hence ∇ · ((a × r)/r) = 0.**

## Check yourself

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

1. ∇ · (φF) equals:
   A. ∇φ × F + φ(∇ × F)
   B. φ∇ · ∇F
   C. ∇φ · ∇F
   D. ∇φ · F + φ(∇ · F)
   Answer: D. With φ = 1/r, F = a × r. (Key point: Product rule)

2. ∇(1/r) equals:
   A. r/r³
   B. −r/r²
   C. −r/r³
   D. −1/r²
   Answer: C. −(1/r²)(r/r). (Key point: ∇(1/r))

3. ∇ · (a × r) equals:
   A. 2a
   B. 0
   C. 3
   D. a
   Answer: B. ∇ × a = 0 and ∇ × r = 0. (Key point: Divergence of a × r)

4. Why is r · (a × r) = 0?
   A. A triple product with a repeated vector vanishes
   B. a = 0
   C. r = 0
   D. Because a ∥ r
   Answer: A. r is perpendicular to a × r. (Key point: Final zero)

## Close variants set in other exams

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