---
title: "Transform vector to cylindrical coordinates"
subject: "Electromagnetism"
chapter: 1
key: "semester_3/electromagnetism:transform-vector-cylindrical-yxz"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/transform-vector-cylindrical-yxz
site: BCT Engineering
---

# Transform vector to cylindrical coordinates

Electromagnetism · Chapter 1: Introduction

- Asked in: 2070 Chaitra, 5 marks
- Video: [EM-Intro Skill 1-07: Convert a vector to cylindrical/spherical coordinates](https://www.youtube.com/watch?v=-Bg-QgWHENc) — Benjamin Yang
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/transform-vector-cylindrical-yxz
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-1

## Question

Transform the vector A = y a_x + x a_y + z a_z into cylindrical co-ordinates at a point P(2, 45°, 5).

## Short answer

At P(2,45°,5), the cylindrical components are A=2 a_ρ + 5 a_z.

## Answer

**Given:** A = y a_x + x a_y + z a_z at P(ρ = 2, φ = 45°, z = 5). Find A_ρ, A_φ, A_z.

#### Method

Cartesian to cylindrical transformation:

```text
A_ρ = A_x cosφ + A_y sinφ
A_φ = −A_x sinφ + A_y cosφ
A_z = A_z
```

Here A_x = y = ρ sinφ, A_y = x = ρ cosφ, A_z = z.

#### Components in terms of ρ, φ, z

```text
A_ρ = ρ sinφ cosφ + ρ cosφ sinφ = 2ρ sinφ cosφ = ρ sin2φ
A_φ = −ρ sin²φ + ρ cos²φ = ρ cos2φ
A_z = z
```

**A = ρ sin2φ a_ρ + ρ cos2φ a_φ + z a_z**

#### Value at P (ρ = 2, φ = 45°, z = 5)

```text
A_ρ = 2 sin90° = 2
A_φ = 2 cos90° = 0
A_z = 5
```

(Check with Cartesian values: x = y = 2 cos45° = 1.414; A = 1.414 a_x + 1.414 a_y + 5 a_z. Then A_ρ = 1.414(0.7071) + 1.414(0.7071) = 2, A_φ = −1.414(0.7071) + 1.414(0.7071) = 0. ✓)

**Answer: A = 2 a_ρ + 5 a_z** at P

## Check yourself

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

1. What is A = y a_x + x a_y + z a_z in cylindrical form?
   A. ρ cos2φ a_ρ + ρ sin2φ a_φ + z a_z
   B. ρ sin2φ a_ρ + ρ cos2φ a_φ + z a_z
   C. ρ a_ρ + z a_z
   D. -ρ a_φ + z a_z
   Answer: B. A_ρ = ρ sinφ cosφ + ρ cosφ sinφ = ρ sin2φ and A_φ = -ρ sin²φ + ρ cos²φ = ρ cos2φ. (Key point: General transformed form)

2. At P(2, 45°, 5), what are A_ρ and A_φ?
   A. A_ρ = 0, A_φ = 2
   B. A_ρ = 2, A_φ = 0
   C. A_ρ = 1.414, A_φ = 1.414
   D. A_ρ = 2, A_φ = 2
   Answer: B. sin90° = 1 and cos90° = 0, so A_ρ = 2 and A_φ = 0. (Key point: Value at P)

3. Which substitution is used for A_x and A_y before transforming?
   A. A_x = ρ cosφ, A_y = ρ sinφ
   B. A_x = y = ρ sinφ, A_y = x = ρ cosφ
   C. A_x = x, A_y = y
   D. A_x = ρ, A_y = φ
   Answer: B. A_x = y = ρ sinφ and A_y = x = ρ cosφ, since the field components are swapped coordinates. (Key point: Substitution used)

## Close variants set in other exams

- [Transform vector B to cylindrical](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Atransform-vector-b-cylindrical) — Electromagnetism, chapter 1, asked 2×
- [Transform vector to spherical coordinates](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Atransform-vector-spherical-point) — Electromagnetism, chapter 1, asked 2×

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