---
title: "Vector field to cylindrical at point"
subject: "Electromagnetism"
chapter: 1
key: "semester_3/electromagnetism:vector-field-transform-cylindrical"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/vector-field-transform-cylindrical
site: BCT Engineering
---

# Vector field to cylindrical at point

Electromagnetism · Chapter 1: Introduction

- Asked in: 2073 Shrawan, 2+3 marks
- Video: [Video-9|Vector Conversion Cartesian to Cylindrical|Engineering Electromagnetic|Engineering Mentor](https://www.youtube.com/watch?v=09q0VWziM6c) — Engineering Mentor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/vector-field-transform-cylindrical
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-1

## Question

Define a vector field. A field vector is given by A = [1/√(x² + y² + z²)] (x a_x + y a_y + z a_z). Transform this vector into cylindrical coordinate system at point (2, 30°, 6).

## Short answer

A vector field assigns a vector to each point in space. At P(2,30°,6), the cylindrical components are A=0.316 a_ρ + 0.949 a_z.

## Answer

**Vector field:** A vector field is a function that assigns a vector (magnitude and direction) to every point in a region of space. Examples: electric field E(x, y, z), velocity of flowing water, gravitational field. (A scalar field assigns only a number, such as temperature.)

#### Transformation

**Given:** A = (x a_x + y a_y + z a_z) / √(x² + y² + z²) at the point (2, 30°, 6), taken as cylindrical coordinates (ρ = 2, φ = 30°, z = 6).

Cartesian position of the point:

```text
x = ρ cosφ = 2 cos30° = 1.732,   y = ρ sinφ = 2 sin30° = 1,   z = 6
r = √(x² + y² + z²) = √(3 + 1 + 36) = √40 = 6.325
```

Cartesian components of A:

```text
A_x = x/r = 0.2739,   A_y = y/r = 0.1581,   A_z = z/r = 0.9487
```

Cartesian to cylindrical:

```text
A_ρ = A_x cosφ + A_y sinφ = 0.2739(0.8660) + 0.1581(0.5) = 0.2372 + 0.0791 = 0.3162
A_φ = −A_x sinφ + A_y cosφ = −0.2739(0.5) + 0.1581(0.8660) = −0.1369 + 0.1369 = 0
A_z = 0.9487
```

**Check:** A is a unit vector (it points radially outward from the origin): 0.3162² + 0.9487² = 0.1 + 0.9 = 1 ✓. In general A_ρ = ρ/√(ρ² + z²) = 2/√40 and A_z = z/√(ρ² + z²) = 6/√40.

**Answer: A = 0.316 a_ρ + 0.949 a_z**

## Check yourself

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

1. What is a vector field?
   A. A function assigning a single number to each point
   B. A function assigning a vector (magnitude and direction) to every point in a region
   C. A vector with a fixed direction
   D. A line integral of a vector
   Answer: B. A vector field gives a vector at every point; a scalar field gives only a number, like temperature. (Key point: Vector field definition)

2. The point (2, 30°, 6) is treated as what, and what are its Cartesian coordinates?
   A. Spherical, x = 1.73
   B. Cylindrical (rho = 2, phi = 30°, z = 6): x = 1.732, y = 1, z = 6
   C. Cartesian (2, 30, 6)
   D. Cylindrical with x = 2, y = 0
   Answer: B. The question gives cylindrical coordinates, so x = 2cos30° = 1.732, y = 2sin30° = 1, z = 6. (Key point: Point interpretation)

3. What is A in cylindrical components at this point?
   A. 0.949 a_ρ + 0.316 a_z
   B. 0.316 a_ρ + 0.949 a_z, with A_φ = 0
   C. 0.316 a_ρ + 0.316 a_φ + 0.949 a_z
   D. 0.274 a_ρ + 0.158 a_φ + 0.949 a_z
   Answer: B. A = r-hat is radial outward, A_ρ = ρ/√(ρ²+z²) = 2/√40 = 0.316, A_z = 6/√40 = 0.949, and A_φ = 0. (Key point: Result)

## Related in Electromagnetism

- [Transform vector to cylindrical coordinates](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Atransform-vector-cylindrical-yxz) — Electromagnetism, chapter 1, asked 3×
- [Transform vector B to cylindrical](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Atransform-vector-b-cylindrical) — Electromagnetism, chapter 1, asked 2×
- [Cylindrical vector to spherical](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Acylindrical-vector-to-spherical-rho-sinphi) — Electromagnetism, chapter 1
- [Cartesian vector to cylindrical](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Acartesian-vector-to-cylindrical-xxy) — Electromagnetism, chapter 1
- [Vector between two points in cylindrical](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Avector-a-to-b-cylindrical) — Electromagnetism, chapter 1
- [Express a_x field in cylindrical, spherical](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aexpress-ax-cylindrical-spherical) — Electromagnetism, chapter 1

## Related in other subjects

- [Force-couple system at A; couple is free vector](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Aforce-couple-system-point-a-box-couple-free-vector) — Applied Mechanics, chapter 3
- [3D world to viewing coordinate matrix](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aworld-to-viewing-3d-matrix) — Computer Graphics, chapter 4, asked 2×
- [Shunt motor speed with 30 Ω field resistance added](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Ashunt-motor-220v-30ohm-field-torque) — Electrical Machines I, chapter 4, asked 5×
- [Cartesian, cylindrical, spherical conversion](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Acoordinate-systems-conversion) — Computer Aided Design, chapter 2