---
title: "Prove i×(a×i)+j×(a×j)+k×(a×k)=2a"
subject: "Mathematics"
chapter: 5
key: "semester_2/mathematics:prove-i-i-j-j"
source: https://www.bctengineer.com/q/semester_2/mathematics/prove-i-i-j-j
site: BCT Engineering
---

# Prove i×(a×i)+j×(a×j)+k×(a×k)=2a

Mathematics · Chapter 5: Vector Algebra and Calculus

- Asked in: 2071 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/prove-i-i-j-j
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-5

## Question

Prove that i×(a×i) + j×(a×j) + k×(a×k) = 2a, where i, j, k are mutually perpendicular unit vectors along the coordinate axes.

## Short answer

Use i×(a×i)=a-(a·i)i and analogous for j, k. Summing gives 3a - (a₁i+a₂j+a₃k) = 2a.

## Answer

Use the vector triple product A × (B × C) = B(A · C) − C(A · B).

#### Each term
With A = i, B = a, C = i:

i × (a × i) = a(i · i) − i(i · a) = a − (a₁) i, since i · i = 1 and i · a = a₁ (the x-component of a).

Similarly:
```text
j × (a × j) = a − (a₂) j
k × (a × k) = a − (a₃) k
```

#### Add
```text
i×(a×i) + j×(a×j) + k×(a×k) = 3a − (a₁i + a₂j + a₃k)
                              = 3a − a
                              = 2a
```

**Hence i × (a × i) + j × (a × j) + k × (a × k) = 2a.**

## Check yourself

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

1. A × (B × C) equals:
   A. B(A·C) − C(A·B)
   B. A(B·C) − C(A·B)
   C. (A·B)C − (A·C)B
   D. B × (A × C)
   Answer: A. BAC-CAB rule. (Key point: Identity)

2. i × (a × i) equals:
   A. a + a1 i
   B. a1 i
   C. 0
   D. a − a1 i
   Answer: D. a(i·i) − i(i·a). (Key point: One term)

3. Adding the three terms gives:
   A. 3a + a
   B. a
   C. 3a − a = 2a
   D. 0
   Answer: C. a1 i + a2 j + a3 k = a. (Key point: Sum)

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