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Change from rectangular to cylindrical coordinates. (let r ≥ 0 and 0 ≤ θ ≤ 2π.) (a) (−9, 9, 9)

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 $(2\sqrt{2}, \dfrac{3 \pi}{4}, 2)$.

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