This question aims to **understand** the rectangular coordinates and **cylindrical** coordinates. Further, it explains how to **convert** from one **coordinate** system into another.

A **rectangular** coordinate system in a plane is a **coordinate** scheme that **identifies** each point **distinctively** by a pair of numerical **coordinates**, which are the signed **lengths** to the point from two bounded **perpendicular** oriented lines, **calculated** in a similar unit of **length.** Each concern **coordinate** line is named a **coordinate** axis or just an axis of the **scheme;** the place where they **intersect** is the origin, and the summoned pair is $(0,0)$.

The **coordinates** can also be described as the situations of the **perpendicular** projections of the pinpoint onto the two axes, defined as signed lengths from the origin. One can utilize the **identical** principle to determine the location of any point in a **three-dimensional** area by three **Rectangular** coordinates, its signed lengths to three mutually vertical planes. In broad, the point in an **n-dimensional** Euclidean space for any dimension $n$ is defined by the $n$ **Rectangular** coordinates. These coordinates are identical, up to sign, to distances from the **juncture** to $n$ mutually abrupt **hyperplanes.**

A **cylindrical** coordinate technique is a **three-dimensional** coordinate scheme that **identifies** point **locations** by the distance from a **selected concerned** axis, the path from the axis comparative to a reference direction chosen (axis $A$), and the span from a selected **considered** plane perpendicular to the axis. The last distance is offered as a **positive** or **negative** numeral relying on that side of the **considered** plane meets the point.

The **origin** of the **scheme** is the end where all **three** coordinates can be **assigned** as zero. This is the **meeting** point between the **considered** plane and the axis. The axis is **variously** named the **cylindrical** axis to distinguish it from the **polar** axis, which is the **beam** that lies in the **considered** plane, **initiating** at the origin and directing in the **reference** path. Other **approaches** perpendicular to the **cylindrical** axis are named **radial** lines.

## Expert Answer

**Rectangular** coordinate is given as $(-9,9,9)$.

The formula for a **cylindrical** coordinate is given by:

\[ r = \sqrt{x^2 + y^2}\]

**Inserting** the values:

\[ r = \sqrt{(-9)^2 + (9)^2} \]

\[ r = \sqrt{81 + 81} \]

\[ r = \sqrt{81 + 81} \]

\[ r = 12.72 \]

\[ \theta = \tan^{-1} \left( \dfrac{y}{x} \right) \]

\[ \theta = \tan^{-1} \left( \dfrac{9}{-9} \right) \]

\[ \theta = \tan^{-1} (-1) \]

\[ \theta = \dfrac{3 \pi}{4} \]

\[ z = z= 9\]

## Numerical Results

**Rectangular** coordinate $(-9,9,9)$ to **cylindrical** coordinate is $(12.72, \dfrac{3 \pi}{4}, 9)$.

## Example

Change **Rectangular** coordinate $(-2,2,2)$ to **cylindrical** coordinate.

Rectangular coordinate is given as $(-2,2,2)$.

The **formula** for finding a **cylindrical** coordinate is provided:

\[ r= \sqrt{x^2+y^2}\]

**Inserting** the values:

\[ r = \sqrt{(-2)^2 + (2)^2} \]

\[ r = \sqrt{4 + 4} \]

\[r=\sqrt{8}\]

\[r=2\sqrt{2}\]

\[\theta=\tan^{-1}\left(\dfrac{y}{x}\right)\]

\[\theta=\tan^{-1}\left(\dfrac{2}{-2}\right)\]

\[\theta= \tan^{-1}(-1)\]

\[ \theta = \dfrac{3 \pi}{4} \]

\[ z = z= 2\]

Rectangular coordinate $(-2,2,2)$ to cylindrical coordinate is $(2\sqrt{2}, \dfrac{3 \pi}{4}, 2)$.