Question 1

1. (1)  Figure out the Cartesian coordinate of (r, θ) = (2, 53 π), and draw it in the Cartesian coordinate system on the plane. √√
2. (2)  Write down the cylindrical coordinate of (x, y, z) = (− 2,

2, −3).
(3) Calculate the Cartesian coordinate of (ρ, θ, φ) = (3, 5π , 2π ), which is written under the

spherical coordinates.

Question 2

Sketch the curve r = 2sinθ written under polar coordinates, and write down its equation using Cartesian coordinates.

Question 3

√ Consider the sphere in dimension 3 centered at (−1, 2, −3), whose radius is 2.

1. (1)  Write down its equation under Cartesian coordinates.
2. (2)  Write down its equation under spherical coordinates. (Hint: plug in the formulas relating Cartesian coordinates and spherical coordinates.)

Question 4

Consider three points P = (1,0,−1),Q = (0,2,1),R = (1,1,1). −→ −→ −→

(1) Figure out the vectors P Q, QR, P R.
−→ −→ −→

(2) Computethelengths(magnitude)ofPQ,QR,PR,anddeterminetheunitvectorsalong the same direction.

Question 5

Given two vectors ⃗a = (2s − t, t + s), ⃗b = (2t + 1, 2 − s), figure out all s and t such that 2⃗a −⃗b = 0.