---
title: "Vector P to Q in three systems"
subject: "Electromagnetism"
chapter: 1
key: "semester_3/electromagnetism:vector-p-to-q-three-systems"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/vector-p-to-q-three-systems
site: BCT Engineering
---

# Vector P to Q in three systems

Electromagnetism · Chapter 1: Introduction

- Asked in: 2069 Chaitra, 5 marks
- Video: [16. Numerical Problem on Transformation of Vectors in Coordinate Systems in Electromagnetic Theory](https://www.youtube.com/watch?v=zPYXGpPLMtM) — Electrical Tutorial
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/vector-p-to-q-three-systems
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-1

## Question

Given a point P(−3, 4, 5), express the vector that extends from P to Q(2, 0, −1) in (a) rectangular co-ordinates (b) cylindrical co-ordinates (c) spherical co-ordinates.

## Short answer

Rectangular: 5 a_x −4 a_y −6 a_z. Cylindrical at P: −6.2 a_ρ −1.6 a_φ −6 a_z. Spherical at P: −8.63 a_r −0.141 a_θ −1.6 a_φ.

## Answer

**Given:** P(−3, 4, 5), Q(2, 0, −1). The vector from P to Q is expressed in each system at P.

#### (a) Rectangular

```text
R_PQ = Q − P = (2 + 3) a_x + (0 − 4) a_y + (−1 − 5) a_z = 5 a_x − 4 a_y − 6 a_z
|R_PQ| = √(25 + 16 + 36) = √77 = 8.775
```

#### (b) Cylindrical (at P)

For P: ρ = √(9 + 16) = 5, φ = tan⁻¹(4/−3) = 126.87° (second quadrant), so cosφ = −0.6, sinφ = 0.8.

```text
R_ρ = R_x cosφ + R_y sinφ = 5(−0.6) + (−4)(0.8) = −3 − 3.2 = −6.2
R_φ = −R_x sinφ + R_y cosφ = −5(0.8) + (−4)(−0.6) = −4 + 2.4 = −1.6
R_z = −6
```

**R_PQ = −6.2 a_ρ − 1.6 a_φ − 6 a_z**

#### (c) Spherical (at P)

For P: r = √(9 + 16 + 25) = √50 = 7.071, θ = cos⁻¹(5/7.071) = 45°, φ = 126.87°. So sinθ = cosθ = 0.7071.

```text
R_r = R_x sinθ cosφ + R_y sinθ sinφ + R_z cosθ
    = 5(0.7071)(−0.6) + (−4)(0.7071)(0.8) + (−6)(0.7071)
    = −2.121 − 2.263 − 4.243 = −8.627

R_θ = R_x cosθ cosφ + R_y cosθ sinφ − R_z sinθ
    = 5(0.7071)(−0.6) + (−4)(0.7071)(0.8) − (−6)(0.7071)
    = −2.121 − 2.263 + 4.243 = −0.141

R_φ = −R_x sinφ + R_y cosφ = −1.6
```

**R_PQ = −8.627 a_r − 0.141 a_θ − 1.6 a_φ**

**Check:** In all systems the magnitude is the same. Cylindrical: 38.44 + 2.56 + 36 = 77 ✓. Spherical: 74.43 + 0.02 + 2.56 = 77 ✓.

## Check yourself

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

1. What is R_PQ in rectangular coordinates?
   A. -5 a_x + 4 a_y + 6 a_z
   B. 5 a_x - 4 a_y - 6 a_z
   C. -1 a_x + 4 a_y + 4 a_z
   D. 5 a_x + 4 a_y - 6 a_z
   Answer: B. Q - P = (2+3, 0-4, -1-5) = (5, -4, -6). (Key point: Rectangular vector)

2. At P(-3, 4, 5), phi lies in which quadrant, and what are cosφ and sinφ?
   A. First: 0.6, 0.8
   B. Second: cosφ = -0.6, sinφ = 0.8 (phi = 126.87°)
   C. Third: -0.6, -0.8
   D. Fourth: 0.6, -0.8
   Answer: B. x < 0 and y > 0 put P in the second quadrant, with rho = 5, so cosφ = -3/5 and sinφ = 4/5. (Key point: Angle at P)

3. What is R_PQ in cylindrical components at P?
   A. -6.2 a_ρ - 1.6 a_φ - 6 a_z
   B. 6.2 a_ρ + 1.6 a_φ - 6 a_z
   C. -1.6 a_ρ - 6.2 a_φ - 6 a_z
   D. 5 a_ρ - 4 a_φ - 6 a_z
   Answer: A. R_ρ = 5(-0.6) + (-4)(0.8) = -6.2 and R_φ = -5(0.8) + (-4)(-0.6) = -1.6; R_z = -6. (Key point: Cylindrical result)

4. How can the cylindrical and spherical results be checked?
   A. The components must equal the Cartesian ones
   B. The sum of squares must equal 77, the same magnitude squared in every system
   C. The components must sum to zero
   D. r must equal rho
   Answer: B. A coordinate change does not change the vector's length, so sum of squares stays 25 + 16 + 36 = 77. (Key point: Magnitude check)

## Related in Electromagnetism

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- [Express a_x field in cylindrical, spherical](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aexpress-ax-cylindrical-spherical) — Electromagnetism, chapter 1
- [Cylindrical vector to spherical](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Acylindrical-vector-to-spherical-rho-sinphi) — Electromagnetism, chapter 1
- [Point transformation rectangular-spherical](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Acoordinate-point-transform-spherical) — 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
- [Vector field to cylindrical at point](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Avector-field-transform-cylindrical) — Electromagnetism, chapter 1

## Related in other subjects

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- [Cylindrical shell with spherical ends: volume change](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Acyl-shell-spherical-ends-volume-change) — Strength of Materials, chapter 5