---
title: "Vector between two points in cylindrical"
subject: "Electromagnetism"
chapter: 1
key: "semester_3/electromagnetism:vector-a-to-b-cylindrical"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/vector-a-to-b-cylindrical
site: BCT Engineering
---

# Vector between two points in cylindrical

Electromagnetism · Chapter 1: Introduction

- Asked in: 2075 Chaitra, 1+4 marks
- Video: [Cylindrical Coordinate System: Convert Cartesian Point to Cylindrical Point and Vice Versa](https://www.youtube.com/watch?v=OeENE7c9OaA) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/vector-a-to-b-cylindrical
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-1

## Question

Find the vector that extends from A(−3, −4, 6) to B(−5, 2, −8) and express it in cylindrical co-ordinate system.

## Short answer

R_AB = B−A = −2 a_x + 6 a_y − 14 a_z. At A, ρ=5, φ=233.13°. Transform: R_ρ = −3.6, R_φ = −5.2, R_z = −14. Hence R_AB = −3.6 a_ρ − 5.2 a_φ − 14 a_z.

## Answer

**Given:** A(−3, −4, 6), B(−5, 2, −8). The vector must be written in cylindrical components. A vector in cylindrical coordinates depends on the point where it is located, so we take its location as the starting point A.

#### Step 1: Vector from A to B (Cartesian)

```text
R_AB = B − A = (−5 + 3) a_x + (2 + 4) a_y + (−8 − 6) a_z
     = −2 a_x + 6 a_y − 14 a_z
```

#### Step 2: Cylindrical coordinates of A

```text
ρ = √(3² + 4²) = 5
φ = tan⁻¹(−4/−3): third quadrant → φ = 180° + 53.13° = 233.13° (−126.87°)
cosφ = −3/5 = −0.6,  sinφ = −4/5 = −0.8
```

#### Step 3: Transform (Cartesian to cylindrical)

```text
R_ρ = R_x cosφ + R_y sinφ = (−2)(−0.6) + (6)(−0.8) = 1.2 − 4.8 = −3.6
R_φ = −R_x sinφ + R_y cosφ = −(−2)(−0.8) + (6)(−0.6) = −1.6 − 3.6 = −5.2
R_z = −14
```

**Check:** (−3.6)² + (−5.2)² + (−14)² = 12.96 + 27.04 + 196 = 236 = (−2)² + 6² + (−14)² ✓

**Answer: R_AB = −3.6 a_ρ − 5.2 a_φ − 14 a_z** (components valid at A, where φ = 233.13°). Its magnitude is √236 = 15.36.

## Check yourself

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

1. What is R_AB for A(-3,-4,6) and B(-5,2,-8) in Cartesian form?
   A. 2 a_x - 6 a_y + 14 a_z
   B. -8 a_x - 2 a_y - 2 a_z
   C. -2 a_x + 6 a_y - 14 a_z
   D. -2 a_x - 6 a_y - 14 a_z
   Answer: C. R_AB = B - A = (-5+3, 2+4, -8-6) = (-2, 6, -14). (Key point: Vector from A to B)

2. What are cosφ and sinφ at A(-3,-4,6)?
   A. cosφ = 0.6, sinφ = 0.8
   B. cosφ = -0.6, sinφ = 0.8
   C. cosφ = 0.8, sinφ = 0.6
   D. cosφ = -0.6, sinφ = -0.8
   Answer: D. A has x, y both negative (third quadrant) and rho = 5, so cosφ = -3/5 and sinφ = -4/5. (Key point: Quadrant of phi at A)

3. Why must the starting point A be used to get the cylindrical components?
   A. Cylindrical unit vectors a_ρ and a_φ change direction with position, so components depend on where the vector is located
   B. Cylindrical vectors only exist at the origin
   C. Because z must be zero
   D. Because B has a larger rho
   Answer: A. a_ρ and a_φ point differently at different places, so the same vector has different cylindrical components at different points. (Key point: Location dependence)

4. What is R_AB in cylindrical components at A?
   A. -3.6 a_ρ - 5.2 a_φ - 14 a_z
   B. 3.6 a_ρ + 5.2 a_φ - 14 a_z
   C. -5.2 a_ρ - 3.6 a_φ - 14 a_z
   D. -1.2 a_ρ - 4.8 a_φ - 14 a_z
   Answer: A. R_ρ = (-2)(-0.6)+6(-0.8) = -3.6 and R_φ = -(-2)(-0.8)+6(-0.6) = -5.2; R_z = -14. (Key point: Cylindrical result)

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