---
title: "Scalar triple product of gradients is zero"
subject: "Mathematics"
chapter: 5
key: "semester_2/mathematics:scalar-triple-product-gradients-zero"
source: https://www.bctengineer.com/q/semester_2/mathematics/scalar-triple-product-gradients-zero
site: BCT Engineering
---

# Scalar triple product of gradients is zero

Mathematics · Chapter 5: Vector Algebra and Calculus

- Asked in: 2068 Bhadra, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/scalar-triple-product-gradients-zero
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-5

## Question

If u = x+y+z, v = x²+y²+z² and w = xy+yz+zx, show that [grad u, grad v, grad w] = 0.

## Short answer

grad u = (1,1,1), grad v = (2x,2y,2z), grad w = (y+z, z+x, x+y). The determinant is 2|1 1 1; x y z; y+z z+x x+y|. Adding row 2 to row 3 makes row 3 proportional to row 1, so the determinant is zero, showing [grad u, grad v, grad w] = 0.

## Answer

u = x + y + z, v = x² + y² + z², w = xy + yz + zx.

#### Gradients
```text
grad u = (1, 1, 1)
grad v = (2x, 2y, 2z)
grad w = (y + z, z + x, x + y)
```

#### Scalar triple product (determinant)
```text
                  |  1      1      1    |
[grad u grad v grad w] = |  2x     2y     2z   |
                  | y+z    z+x    x+y  |
```
Take 2 out of the second row:

```text
= 2 | 1     1     1   |
    | x     y     z   |
    | y+z   z+x   x+y |
```
Now add the second row to the third row (R₃ → R₃ + R₂):

third row = (y + z + x, z + x + y, x + y + z) = (x + y + z)(1, 1, 1),

which is a multiple of the first row. A determinant with two proportional rows is zero.

**Hence [grad u, grad v, grad w] = 0.**

This shows u, v, w are functionally dependent: indeed v = u² − 2w.

## Check yourself

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

1. grad w for w = xy + yz + zx is:
   A. (x, y, z)
   B. (y, z, x)
   C. (y + z, z + x, x + y)
   D. (2x, 2y, 2z)
   Answer: C. Partial derivatives. (Key point: Gradients)

2. R3 → R3 + R2 turns the third row into:
   A. (0, 0, 0)
   B. (x + y + z)(1, 1, 1)
   C. (2x, 2y, 2z)
   D. (x, y, z)
   Answer: B. (y+z)+x etc. (Key point: Row operation)

3. The determinant is zero because:
   A. two rows are proportional
   B. it has a zero row
   C. it is diagonal
   D. it is symmetric
   Answer: A. Row 3 is a multiple of row 1. (Key point: Zero determinant)

4. The functions satisfy:
   A. v = u² + 2w
   B. v = u + w
   C. v = 2uw
   D. v = u² − 2w
   Answer: D. u² = v + 2w. (Key point: Dependence)

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