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Calculus 3 : 3-Dimensional Space

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Example Questions

Example Question #215 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (125,18^{\circ},26).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(82.54,-93.87,26)

(118.88,38.63,26)

(38.63,118.88,26)

(-93.87,82.54,26)

Correct answer:

(118.88,38.63,26)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=125cos(18^{\circ})=118.88\\&y=125sin(18^{\circ})=38.63\\&z=26\end{align*}$

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Example Question #216 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (138,93^{\circ},-64).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(43.81,-130.86,-64)

(-7.22,137.81,-64)

(137.81,-7.22,-64)

(-130.86,43.81,-64)

Correct answer:

(-7.22,137.81,-64)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=138cos(93^{\circ})=-7.22\\&y=138sin(93^{\circ})=137.81\\&z=-64\end{align*}$

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Example Question #301 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (114,24^{\circ},-36).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(48.36,-103.24,-36)

(-103.24,48.36,-36)

(46.37,104.14,-36)

(104.14,46.37,-36)

Correct answer:

(104.14,46.37,-36)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=114cos(24^{\circ})=104.14\\&y=114sin(24^{\circ})=46.37\\&z=-36\end{align*}$

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Example Question #218 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (12,11^{\circ},-134).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(11.78,2.29,-134)

(0.05,-12,-134)

(-12,0.05,-134)

(2.29,11.78,-134)

Correct answer:

(11.78,2.29,-134)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=12cos(11^{\circ})=11.78\\&y=12sin(11^{\circ})=2.29\\&z=-134\end{align*}$

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Example Question #219 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (117,-134^{\circ},131).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-54.27,-103.65,131)

(-84.16,-81.28,131)

(-103.65,-54.27,131)

(-81.28,-84.16,131)

Correct answer:

(-81.28,-84.16,131)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=117cos(-134^{\circ})=-81.28\\&y=117sin(-134^{\circ})=-84.16\\&z=131\end{align*}$

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Example Question #220 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (86,-176^{\circ},-9).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-85.79,-6,-9)

(-6.08,85.78,-9)

(85.78,-6.08,-9)

(-6,-85.79,-9)

Correct answer:

(-85.79,-6,-9)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=86cos(-176^{\circ})=-85.79\\&y=86sin(-176^{\circ})=-6\\&z=-9\end{align*}$

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Example Question #221 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (51,106^{\circ},-102).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-14.06,49.02,-102)

(49.02,-14.06,-102)

(-37.08,35.01,-102)

(35.01,-37.08,-102)

Correct answer:

(-14.06,49.02,-102)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=51cos(106^{\circ})=-14.06\\&y=51sin(106^{\circ})=49.02\\&z=-102\end{align*}$

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Example Question #222 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (47,-121^{\circ},9).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-46.94,-2.29,9)

(-2.29,-46.94,9)

(-40.29,-24.21,9)

(-24.21,-40.29,9)

Correct answer:

(-24.21,-40.29,9)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=47cos(-121^{\circ})=-24.21\\&y=47sin(-121^{\circ})=-40.29\\&z=9\end{align*}$

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Example Question #1971 : Calculus 3

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (91,56^{\circ},-71).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(77.64,-47.46,-71)

(50.89,75.44,-71)

(-47.46,77.64,-71)

(75.44,50.89,-71)

Correct answer:

(50.89,75.44,-71)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=91cos(56^{\circ})=50.89\\&y=91sin(56^{\circ})=75.44\\&z=-71\end{align*}$

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Example Question #1972 : Calculus 3

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (104,-18^{\circ},75).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(98.91,-32.14,75)

(-32.14,98.91,75)

(68.67,78.1,75)

(78.1,68.67,75)

Correct answer:

(98.91,-32.14,75)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=104cos(-18^{\circ})=98.91\\&y=104sin(-18^{\circ})=-32.14\\&z=75\end{align*}$

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Calculus 3 : 3-Dimensional Space

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #215 : Cylindrical Coordinates

Example Question #216 : Cylindrical Coordinates

Example Question #301 : 3 Dimensional Space

Example Question #218 : Cylindrical Coordinates

Example Question #219 : Cylindrical Coordinates

Example Question #220 : Cylindrical Coordinates

Example Question #221 : Cylindrical Coordinates

Example Question #222 : Cylindrical Coordinates

Example Question #1971 : Calculus 3

Example Question #1972 : Calculus 3

All Calculus 3 Resources

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