---
title: "Cartesian vector to cylindrical"
subject: "Electromagnetism"
chapter: 1
key: "semester_3/electromagnetism:cartesian-vector-to-cylindrical-xxy"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/cartesian-vector-to-cylindrical-xxy
site: BCT Engineering
---

# Cartesian vector to cylindrical

Electromagnetism · Chapter 1: Introduction

- Asked in: 2067 Mangsir, 6 marks
- Video: [Cartesian to Cylindrical Vector Conversion Explained](https://www.youtube.com/watch?v=Ya1S1uqgYP4) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/cartesian-vector-to-cylindrical-xxy
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-1

## Question

Transform A_c = x a_x + xy a_y + a_z at the point (1, 2, 3) in Cartesian co-ordinate system to A_cy in cylindrical co-ordinate system.

## Short answer

At (1,2,3), ρ=√5, φ=63.43°. A_x=1, A_y=2, A_z=1. A_ρ = A_x cosφ + A_y sinφ = 2.236, A_φ = -A_x sinφ + A_y cosφ = 0, A_z=1. So A_cy = 2.236 a_ρ + a_z.

## Answer

**Given:** A_c = x a_x + xy a_y + a_z at the point (1, 2, 3). Find A_cy (cylindrical).

#### Step 1: Cylindrical coordinates of the point

```text
ρ = √(1² + 2²) = √5 = 2.236
φ = tan⁻¹(2/1) = 63.43°        cosφ = 1/√5 = 0.4472,  sinφ = 2/√5 = 0.8944
z = 3
```

#### Step 2: Cartesian components at the point

```text
A_x = x = 1,   A_y = xy = 2,   A_z = 1
```

#### Step 3: Transform

```text
A_ρ = A_x cosφ + A_y sinφ = 1(0.4472) + 2(0.8944) = 0.4472 + 1.7889 = 2.236
A_φ = −A_x sinφ + A_y cosφ = −1(0.8944) + 2(0.4472) = −0.8944 + 0.8944 = 0
A_z = 1
```

**Check:** |A|² = 1 + 4 + 1 = 6, and 2.236² + 0 + 1² = 5 + 1 = 6 ✓

**Answer: A_cy = 2.236 a_ρ + a_z** (that is, √5 a_ρ + a_z) at (1, 2, 3).

General form: A_ρ = ρ cos²φ + ρ² cosφ sin²φ, A_φ = −ρ sinφ cosφ + ρ² sinφ cos²φ, A_z = 1. At ρ = √5 these give the same values as above.

## Check yourself

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

1. What are A_x, A_y, A_z at (1, 2, 3)?
   A. 1, 2, 1
   B. 1, 2, 3
   C. 1, 1, 1
   D. 2, 1, 1
   Answer: A. A_x = x = 1, A_y = xy = 2, and the a_z component is constant 1. (Key point: Cartesian components at point)

2. What are cosφ and sinφ at the point?
   A. 1/√5 = 0.4472 and 2/√5 = 0.8944
   B. 2/√5 and 1/√5
   C. 0.5 and 0.866
   D. 1 and 2
   Answer: A. phi = tan^-1(2/1) = 63.43°, with rho = √5, giving cosφ = 1/√5 and sinφ = 2/√5. (Key point: Angle used)

3. What is A_φ at the point?
   A. 2.236
   B. 0, since -1(0.8944) + 2(0.4472) = 0
   C. -0.894
   D. 1
   Answer: B. A_φ = -A_x sinφ + A_y cosφ = -0.8944 + 0.8944 = 0. (Key point: A_phi)

4. What is A_cy and the check?
   A. √5 a_ρ + a_z; |A|^2 = 6 = 5 + 1
   B. a_ρ + √5 a_z; |A|^2 = 6
   C. 2.236 a_ρ + 0.5 a_φ + a_z
   D. √5 a_ρ + a_z; |A|^2 = 5
   Answer: A. A_ρ = 0.4472 + 1.7889 = 2.236, A_z = 1, and 2.236² + 1 = 6 matches 1 + 4 + 1. (Key point: Result and check)

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