Vector P to Q in three systems

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

(a) Rectangular

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.

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.

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 ✓.