---
title: "Triple product for r = a cos t i + a sin t j + at tanα k"
subject: "Mathematics"
chapter: 5
key: "semester_2/mathematics:triple-product-r-cos-t"
source: https://www.bctengineer.com/q/semester_2/mathematics/triple-product-r-cos-t
site: BCT Engineering
---

# Triple product for r = a cos t i + a sin t j + at tanα k

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/triple-product-r-cos-t
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-5

## Question

If r = a cos t i + a sin t j + at tanα k, find [r dr/dt d²r/dt²].

## Short answer

Compute derivatives: r' = (-a sin t, a cos t, a tan α), r'' = (-a cos t, -a sin t, 0). r'×r'' = a² tan α sin t i - a² tan α cos t j + a² k. Then r·(r'×r'') = a³ t tan α.

## Answer

[Reading: r = a cos t i + a sin t j + a t tan α k, with a and α constants.]

We need [r, dr/dt, d²r/dt²] = r · (dr/dt × d²r/dt²).

#### Derivatives
```text
r   = (a cos t,  a sin t,  a t tan α)
r′  = (−a sin t, a cos t,  a tan α)
r″  = (−a cos t, −a sin t, 0)
```

#### r′ × r″
```text
r′ × r″ = ( a cos t·0 − a tan α·(−a sin t),
            a tan α·(−a cos t) − (−a sin t)·0,
            (−a sin t)(−a sin t) − (a cos t)(−a cos t) )
        = ( a² tan α sin t, −a² tan α cos t, a² )
```

#### Dot with r
```text
r · (r′ × r″) = a cos t · a² tan α sin t − a sin t · a² tan α cos t + a t tan α · a²
              = a³ tan α sin t cos t − a³ tan α sin t cos t + a³ t tan α
              = a³ t tan α
```

**Answer: [r r′ r″] = a³ t tan α.**

## Check yourself

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

1. [r r' r''] means:
   A. r × (r' · r'')
   B. r' · r''
   C. |r||r'||r''|
   D. r · (r' × r'')
   Answer: D. Scalar triple product. (Key point: Meaning)

2. r'' for r = (a cos t, a sin t, at tanα) is:
   A. (−a sin t, a cos t, a tanα)
   B. (a cos t, a sin t, 0)
   C. (−a cos t, −a sin t, 0)
   D. (−a cos t, −a sin t, a tanα)
   Answer: C. Second derivative; the linear z term drops out. (Key point: Derivatives)

3. The z-component of r' × r'' is:
   A. −a²
   B. a²
   C. a² tanα
   D. 0
   Answer: B. a² sin²t + a² cos²t. (Key point: z-component)

4. [r r' r''] equals:
   A. a³ t tanα
   B. a³ tanα
   C. a³ t
   D. 0
   Answer: A. The sin t cos t terms cancel. (Key point: Answer)

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