Question

convert the point from rectangular coordinates to cylindrical coordinates. (9, 3√3, -1) (r, θ, z) = ( )

Explanation:

Step1: Calculate the value of \(r\)

The formula for \(r\) in cylindrical - coordinates when converting from rectangular coordinates \((x,y,z)\) is \(r = \sqrt{x^{2}+y^{2}}\). Given \(x = 9\) and \(y = 3\sqrt{3}\), then \(r=\sqrt{9^{2}+(3\sqrt{3})^{2}}=\sqrt{81 + 27}=\sqrt{108}=6\sqrt{3}\).

Step2: Calculate the value of \(\theta\)

The formula for \(\theta\) is \(\tan\theta=\frac{y}{x}\). Here, \(\tan\theta=\frac{3\sqrt{3}}{9}=\frac{\sqrt{3}}{3}\). Since \(x = 9>0\) and \(y = 3\sqrt{3}>0\), \(\theta=\frac{\pi}{6}\) (in the first - quadrant).

Step3: The \(z\) value remains the same

The \(z\) - coordinate in rectangular coordinates is \(z=-1\), and in cylindrical coordinates, \(z\) remains \(-1\).

Answer:

\((6\sqrt{3},\frac{\pi}{6}, - 1)\)