--- title: "Transform vector to spherical coordinates" subject: "Electromagnetism" chapter: 1 key: "semester_3/electromagnetism:transform-vector-spherical-point" source: https://www.bctengineer.com/q/semester_3/electromagnetism/transform-vector-spherical-point site: BCT Engineering --- # Transform vector to spherical coordinates Electromagnetism · Chapter 1: Introduction - Asked in: 2076 Chaitra, 5 marks - Video: [Vector conversion || Rectangular || Cylindrical || Spherical || by Prof. Niraj Kumar VIT Chennai](https://www.youtube.com/watch?v=I3L22NosQg0) — RF Design Basics - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/electromagnetism/transform-vector-spherical-point - Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-1 ## Question Transform the vector A = 4a_x − 2a_y − 4a_z into spherical co-ordinates at a point P(x = −2, y = −3, z = 4). ## Short answer At P, r = √29 ≈ 5.385, θ = 42.03°, φ = 236.31°. Using transformation matrix: A_r = −3.343, A_θ = 2.266, A_φ = 4.438. Thus A = −3.34 a_r + 2.27 a_θ + 4.44 a_φ. ## Answer **Given:** A = 4a_x − 2a_y − 4a_z at P(x = −2, y = −3, z = 4). Find A_r, A_θ, A_φ. #### Transformation (Cartesian to spherical) ```text | A_r | | sinθ cosφ sinθ sinφ cosθ | | A_x | | A_θ | = | cosθ cosφ cosθ sinφ −sinθ | | A_y | | A_φ | | −sinφ cosφ 0 | | A_z | ``` #### Step 1: Location of P in spherical coordinates ```text r = √(x² + y² + z²) = √(4 + 9 + 16) = √29 = 5.385 θ = cos⁻¹(z/r) = cos⁻¹(4/5.385) = 42.03° φ = tan⁻¹(y/x) : x < 0, y < 0 (third quadrant) → φ = 180° + tan⁻¹(3/2) = 236.31° (= −123.69°) ``` So sinθ = √13/√29 = 0.6695, cosθ = 4/√29 = 0.7428, cosφ = −2/√13 = −0.5547, sinφ = −3/√13 = −0.8321. #### Step 2: Components ```text A_r = A_x sinθ cosφ + A_y sinθ sinφ + A_z cosθ = 4(0.6695)(−0.5547) + (−2)(0.6695)(−0.8321) + (−4)(0.7428) = −1.4856 + 1.1142 − 2.9712 = −3.343 A_θ = A_x cosθ cosφ + A_y cosθ sinφ − A_z sinθ = 4(0.7428)(−0.5547) + (−2)(0.7428)(−0.8321) − (−4)(0.6695) = −1.6481 + 1.2362 + 2.6780 = 2.266 A_φ = −A_x sinφ + A_y cosφ = −4(−0.8321) + (−2)(−0.5547) = 3.3284 + 1.1094 = 4.438 ``` **Check:** |A|² = 16 + 4 + 16 = 36, and (−3.343)² + (2.266)² + (4.438)² = 11.18 + 5.13 + 19.70 = 36.0. ✓ **Answer: A = −3.34 a_r + 2.27 a_θ + 4.44 a_φ** (at P, r = 5.385, θ = 42.03°, φ = 236.3°) ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For P(-2, -3, 4), what are r and theta? A. r = 5.385, theta = 42.03° B. r = 5.385, theta = 47.97° C. r = 29, theta = 42.03° D. r = 3.606, theta = 42.03° Answer: A. r = sqrt(4+9+16) = sqrt(29) = 5.385 and theta = cos^-1(z/r) = cos^-1(4/5.385) = 42.03°. (Key point: Location of the point) 2. Why is phi taken as 236.31° and not 56.31°? A. Because z is positive B. Because theta is less than 90° C. Because x and y are both negative, so the point is in the third quadrant D. Because tan^-1 always adds 180° Answer: C. tan^-1(3/2) = 56.31°, but with x < 0 and y < 0 the point lies in the third quadrant, so phi = 180° + 56.31°. (Key point: Quadrant of phi) 3. Which expression gives A_phi? A. A_x cosφ + A_y sinφ B. -A_x sinφ + A_y cosφ C. A_x sinθ cosφ + A_y sinθ sinφ D. A_x cosθ cosφ + A_y cosθ sinφ Answer: B. The phi row of the transformation matrix is (-sinφ, cosφ, 0), so A_phi = -A_x sinφ + A_y cosφ. (Key point: A_phi formula) 4. Which is the final spherical vector, and how is it checked? A. -3.34 a_r + 2.27 a_θ + 4.44 a_φ; the squares of components add to |A|^2 = 36 B. 3.34 a_r + 2.27 a_θ - 4.44 a_φ; the sum of components equals 6 C. -3.34 a_r - 2.27 a_θ + 4.44 a_φ; the squares add to 6 D. -3.34 a_r + 2.27 a_θ + 4.44 a_φ; the components add to 3.37 Answer: A. A is rotated, not stretched, so the sum of squares must equal 16 + 4 + 16 = 36, which holds for (-3.34, 2.27, 4.44). 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