The coordinates (2, 1, -2) corresponds to: x = 2, y = 1, z = -2, and are to be converted to the cylindrical coordinates in form of (r, θ, z), where:

\(\displaystyle \\ r = \sqrt{x^{2}+y^{2}} \\ \tan(\theta) = \frac{y}{x} \\ z = z\)

So, filling in for x, y, z:

\(\displaystyle \\ r = {\sqrt{2^{2}+1^{2}}} = \sqrt{5} \\ \tan(\theta) = \frac{1}{2} \Rightarrow \theta = \tan^{-1}\frac{1}{2} \\ z = -2\)

Then the cylindrical coordinates are represented as:

\(\displaystyle \left ( \sqrt{5}, \tan^{-1}\frac{1}{2}, -2 \right )\)

The coordinates (0, 3, 4) corresponds to: x = 0, y = 3, z = 4, and are to be converted to the cylindrical coordinates in form of (r, θ, z), where:

\(\displaystyle \\ r = \sqrt{x^{2}+y^{2}} \\ \tan(\theta) = \frac{y}{x} \\ z = z\)

So, filling in for x, y, z:

\(\displaystyle \\ r = {\sqrt{0^{2}+3^{2}}} = \sqrt{9} = 3 \\ \tan(\theta) = \frac{3}{0} \Rightarrow \theta = \frac{1}{2}\pi \\ z = 4\)

Then the cylindrical coordinates are represented as:

\(\displaystyle \left (3, \frac{1}{2}\pi, 4 \right )\)

The coordinates (√2, 1, 1) corresponds to: x = √2, y = 1, z = 1, and are to be converted to the cylindrical coordinates in form of (r, θ, z), where:

\(\displaystyle \\ r = \sqrt{x^{2}+y^{2}} \\ \tan(\theta) = \frac{y}{x} \\ z = z\)

So, filling in for x, y, z:

\(\displaystyle \\ r = {\sqrt{\sqrt{2}^{2}+1^{2}}} = \sqrt{2+1} = \sqrt{3} \\ \tan(\theta) = \frac{1}{\sqrt{2}} \Rightarrow \theta = \tan^{-1}\frac{\sqrt{2}}{2} \\ z = 1\)

Then the cylindrical coordinates are represented as:

\(\displaystyle \left (\sqrt{3}, \tan^{-1}\frac{\sqrt{2}}{2}, 1 \right )\)

The coordinates (3, π/3, -4) corresponds to: r = 3, θ = π/3, z = -4, and are to be converted to the Cartesian coordinates in form of (x, y, z), where:

\(\displaystyle \\ x = r\cos\theta \\ y = r\sin\theta \\ z = z\)

So, filling in for r, θ, z:

\(\displaystyle \\ x = 3\cos{\frac{1}{3}\pi} = 3\cdot \frac{1}{2} = \frac{3}{2} \\ y = 3\sin{\frac{1}{3}\pi} = 3\cdot \frac{\sqrt{3}}{2} = \frac{3\sqrt{2}}{2} \\ z = -4\)

Then the Cartesian coordinates are represented as:

\(\displaystyle \left (\frac{3}{2}, \frac{3\sqrt{3}}{2}, -4 \right)\)

The coordinates (-2, 2, 3) corresponds to: x = -2, y = 2, z = 3, and are to be converted to the cylindrical coordinates in form of (r, θ, z), where:

\(\displaystyle \\ r = \sqrt{x^{2}+y^{2}} \\ \tan(\theta) = \frac{y}{x} \\ z = z\)

So, filling in for x, y, z:

\(\displaystyle \\ r = {\sqrt{\left ( -2\right )^{2}+2^{2}}} = \sqrt{4+4} = \sqrt{8} = 2\sqrt{2} \\ \tan(\theta) = \frac{2}{-2} \Rightarrow \theta = \tan^{-1}\ \left ( -1\right ) \Rightarrow \theta = \frac{3}{4}\pi \\ z = 3\)

Then the cylindrical coordinates are represented as:

\(\displaystyle \left ( 2\sqrt{2}, \frac{3}{4}\pi, 3\right )\)

The coordinates (1, 45°, 1) corresponds to: r = 1, θ = 45°, z = 1, and are to be converted to the Cartesian coordinates in form of (x, y, z), where:

\(\displaystyle \\ x = r\cos\theta \\ y = r\sin\theta \\ z = z\)

So, filling in for r, θ, z:

\(\displaystyle \\ x = 1\cos{45^{\circ}} = 1\cdot \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \\ y = 1\sin{45^{\circ}} = 1\cdot \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \\ z = 1\)

Then the Cartesian coordinates are represented as:

\(\displaystyle \left ( \frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}, 1\right )\)

The coordinates (2, π/2, -4) corresponds to: r = 2, θ = π/2, z = -4, and are to be converted to the Cartesian coordinates in form of (x, y, z), where:

\(\displaystyle \\ x = r\cos\theta \\ y = r\sin\theta \\ z = z\)

So, filling in for r, θ, z:

\(\displaystyle \\ x = 2\cos{\frac{1}{2}\pi} = 2\cdot 0 = 0 \\ y = 2\sin{\frac{1}{2}\pi} = 2\cdot 1 = 2 \\ z = -4\)

Then the Cartesian coordinates are represented as:

\(\displaystyle \left ( 0,2,-4\right )\)