- 2075 Asoj · 3+2 marks
Cylindrical points to cartesian vector
Short answer
Convert A(4,40°,−2) and B(5,−110°,2) to Cartesian: A(3.064,2.571,−2), B(−1.710,−4.698,2). (a) R_AB = −4.77 a_x − 7.27 a_y + 4 a_z. (b) Unit vector from B to A: 0.499 a_x + 0.759 a_y − 0.418 a_z.
Express in cartesian components: (a) the vector at A(ρ = 4, φ = 40°, z = −2) that extends to B(ρ = 5, φ = −110°, z = 2); (b) a unit vector at B directed toward A.
Given: A(ρ = 4, φ = 40°, z = −2), B(ρ = 5, φ = −110°, z = 2).
Step 1: Cartesian coordinates of the points
A: x = 4 cos40° = 3.064, y = 4 sin40° = 2.571, z = −2
B: x = 5 cos(−110°) = −1.710, y = 5 sin(−110°) = −4.698, z = 2
(a) Vector from A to B
R_AB = B − A = (−1.710 − 3.064) a_x + (−4.698 − 2.571) a_y + (2 + 2) a_z
= −4.774 a_x − 7.270 a_y + 4 a_z
Its magnitude is |R_AB| = √(22.79 + 52.84 + 16) = √91.63 = 9.573.
(b) Unit vector at B directed toward A
The direction from B to A is R_BA = A − B = 4.774 a_x + 7.270 a_y − 4 a_z, so
a_BA = R_BA / |R_BA| = (4.774 a_x + 7.270 a_y − 4 a_z) / 9.573
= 0.499 a_x + 0.759 a_y − 0.418 a_z
Check: 0.499² + 0.759² + 0.418² = 0.249 + 0.576 + 0.175 = 1.000 ✓
Answer:
- (a) R_AB = −4.77 a_x − 7.27 a_y + 4 a_z
- (b) a_BA = 0.499 a_x + 0.759 a_y − 0.418 a_z
✓ Answer checked by Claude Opus 5.5 (high effort)
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