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Calculus 3 : Cylindrical Coordinates

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Example Question #241 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (107,-81^{\circ},77).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-105.68,16.74,77)

(16.74,-105.68,77)

(83.11,67.4,77)

(67.4,83.11,77)

Correct answer:

(16.74,-105.68,77)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=107cos(-81^{\circ})=16.74\\&y=107sin(-81^{\circ})=-105.68\\&z=77\end{align*}$

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Example Question #321 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (102,-122^{\circ},47).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-88.41,-50.87,47)

(-86.5,-54.05,47)

(-50.87,-88.41,47)

(-54.05,-86.5,47)

Correct answer:

(-54.05,-86.5,47)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=102cos(-122^{\circ})=-54.05\\&y=102sin(-122^{\circ})=-86.5\\&z=47\end{align*}$

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Example Question #242 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (18,166^{\circ},0).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(8.7,-15.76,0)

(4.35,-17.47,0)

(-15.76,8.7,0)

(-17.47,4.35,0)

Correct answer:

(-17.47,4.35,0)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=18cos(166^{\circ})=-17.47\\&y=18sin(166^{\circ})=4.35\\&z=0\end{align*}$

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Example Question #243 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (52,-100^{\circ},26).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(26.33,44.84,26)

(-9.03,-51.21,26)

(-51.21,-9.03,26)

(44.84,26.33,26)

Correct answer:

(-9.03,-51.21,26)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=52cos(-100^{\circ})=-9.03\\&y=52sin(-100^{\circ})=-51.21\\&z=26\end{align*}$

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Example Question #244 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (113,2^{\circ},-74).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(102.75,-47.02,-74)

(112.93,3.94,-74)

(-47.02,102.75,-74)

(3.94,112.93,-74)

Correct answer:

(112.93,3.94,-74)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=113cos(2^{\circ})=112.93\\&y=113sin(2^{\circ})=3.94\\&z=-74\end{align*}$

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Example Question #245 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (105,166^{\circ},118).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(25.4,-101.88,118)

(-91.92,50.75,118)

(-101.88,25.4,118)

(50.75,-91.92,118)

Correct answer:

(-101.88,25.4,118)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=105cos(166^{\circ})=-101.88\\&y=105sin(166^{\circ})=25.4\\&z=118\end{align*}$

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Example Question #246 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (83,-127^{\circ},-109).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-66.29,-49.95,-109)

(19.29,-80.73,-109)

(-49.95,-66.29,-109)

(-80.73,19.29,-109)

Correct answer:

(-49.95,-66.29,-109)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=83cos(-127^{\circ})=-49.95\\&y=83sin(-127^{\circ})=-66.29\\&z=-109\end{align*}$

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Example Question #247 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (39,-89^{\circ},103).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(0.68,-38.99,103)

(-38.99,0.68,103)

(-33.54,19.9,103)

(19.9,-33.54,103)

Correct answer:

(0.68,-38.99,103)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=39cos(-89^{\circ})=0.68\\&y=39sin(-89^{\circ})=-38.99\\&z=103\end{align*}$

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Example Question #248 : Cylindrical Coordinates

Find a parametric representation of the circle 3x^2 + 9 y ^2 = 27 .

Possible Answers:

$\vec{r}(t)= \left< \cos 3t, \sqrt{3} \sin 3t \right>$

$\vec{r}(t)= \left< \cos t, \sin t \right>$

$\vec{r}(t)= \left< 3 \cos t, \sqrt{3} \sin t \right>$

$\vec{r}(t)= \left< 3 \cos t, \sqrt{3} \tan t \right>$

Correct answer:

$\vec{r}(t)= \left< 3 \cos t, \sqrt{3} \sin t \right>$

Explanation:

We can begin by rewriting the equation for a circle as

$3x^2 + 9y^2 = 27 \Rightarrow \left(\frac{x}{3} \right )^2 + \left(\frac{y}{\sqrt{3}} \right )^2=1$ .

This directly tells us that $x = 3 \cos t, y = \sqrt{3} \sin t$ . This allows us to write our final expression for the parametric representation as

$\vec{r}(t)= \left< 3 \cos t, \sqrt{3} \sin t \right>$

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Example Question #249 : Cylindrical Coordinates

Convert the following vector in Cartesian coordinates into cylindrical coordinates.

(3xy, x^2z, y)

Possible Answers:

$(3r\cos\theta \sin \theta, rz\cos^2 \theta, r\sin \theta )$

$(3r^2\sin\theta \cos\theta, rz\sin^2 \theta, r\cos \theta )$

$(3r^2\sin\theta \cos\theta, r\sin^2 \theta, r\cos \theta )$

$(3r^2\cos\theta \sin \theta, rz\cos^2 \theta, r\sin \theta )$

$(3r^2\sin\theta \cos\theta, r\sin^2 \theta, \cos \theta )$

Correct answer:

$(3r^2\cos\theta \sin \theta, rz\cos^2 \theta, r\sin \theta )$

Explanation:

The conversion from Cartesian to cylindrical coordinates is as follows:

$x=r\cos \theta$

$y=r \sin \theta$

z=z

The three components of the vector then become:

$3xy\rightarrow3r\cos \theta \:r\sin \theta=3r^2\cos\theta \sin\theta$

$x^2z\rightarrow r^2 \cos^2\theta z$

$y\rightarrow r\sin \theta$

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Calculus 3 : Cylindrical Coordinates

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #241 : Cylindrical Coordinates

Example Question #321 : 3 Dimensional Space

Example Question #242 : Cylindrical Coordinates

Example Question #243 : Cylindrical Coordinates

Example Question #244 : Cylindrical Coordinates

Example Question #245 : Cylindrical Coordinates

Example Question #246 : Cylindrical Coordinates

Example Question #247 : Cylindrical Coordinates

Example Question #248 : Cylindrical Coordinates

Example Question #249 : Cylindrical Coordinates

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