Unit vector from A toward B

Given: A(ρ = 5, φ = 70°, z = −3), B(ρ = 2, φ = −30°, z = 1).

Step 1: Convert both points to Cartesian

A: x = 5 cos70° = 1.710,  y = 5 sin70° = 4.698,  z = −3
B: x = 2 cos(−30°) = 1.732,  y = 2 sin(−30°) = −1.000,  z = 1

(a) Unit vector in Cartesian coordinates

R_AB = B − A = (1.732 − 1.710) a_x + (−1.000 − 4.698) a_y + (1 + 3) a_z
     = 0.022 a_x − 5.698 a_y + 4 a_z

|R_AB| = √(0.022² + 5.698² + 4²) = √(0.0005 + 32.47 + 16) = 6.962

a_AB = R_AB / |R_AB| = 0.0032 a_x − 0.8185 a_y + 0.5745 a_z

(b) Unit vector in cylindrical coordinates at A

At A, φ = 70° (cos70° = 0.3420, sin70° = 0.9397). Project the Cartesian components of a_AB onto a_ρ and a_φ at A:

a_ρ component = (x-component) cosφ + (y-component) sinφ
a_φ component = −(x-component) sinφ + (y-component) cosφ

a_ρ component = 0.0032(0.3420) + (−0.8185)(0.9397) = 0.0011 − 0.7692 = −0.768
a_φ component = −0.0032(0.9397) + (−0.8185)(0.3420) = −0.0030 − 0.2799 = −0.283
a_z component = 0.5745

Check: 0.768² + 0.283² + 0.5745² = 0.590 + 0.080 + 0.330 = 1.000 ✓

Answer:

• (a) a_AB = 0.003 a_x − 0.819 a_y + 0.575 a_z
• (b) a_AB = −0.768 a_ρ − 0.283 a_φ + 0.575 a_z (at A)