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Calculus 3 : 3-Dimensional Space

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #1855 : Calculus 3

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (9,101^{\circ},9).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(4.07,8.03,9)

(8.03,4.07,9)

(-1.72,8.83,9)

(8.83,-1.72,9)

Correct answer:

(-1.72,8.83,9)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=9cos(101^{\circ})=-1.72\\&y=9sin(101^{\circ})=8.83\\&z=9\end{align*}$

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Example Question #1856 : Calculus 3

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (71,-59^{\circ},-147).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-45.21,-54.75,-147)

(36.57,-60.86,-147)

(-60.86,36.57,-147)

(-54.75,-45.21,-147)

Correct answer:

(36.57,-60.86,-147)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=71cos(-59^{\circ})=36.57\\&y=71sin(-59^{\circ})=-60.86\\&z=-147\end{align*}$

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Example Question #1857 : Calculus 3

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (25,-68^{\circ},89).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(9.37,-23.18,89)

(22.45,11,89)

(11,22.45,89)

(-23.18,9.37,89)

Correct answer:

(9.37,-23.18,89)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=25cos(-68^{\circ})=9.37\\&y=25sin(-68^{\circ})=-23.18\\&z=89\end{align*}$

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Example Question #1858 : Calculus 3

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (80,37^{\circ},-101).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(63.89,48.15,-101)

(48.15,63.89,-101)

(61.23,-51.48,-101)

(-51.48,61.23,-101)

Correct answer:

(63.89,48.15,-101)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=80cos(37^{\circ})=63.89\\&y=80sin(37^{\circ})=48.15\\&z=-101\end{align*}$

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Example Question #1861 : Calculus 3

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (40,68^{\circ},46).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(14.98,37.09,46)

(37.09,14.98,46)

(17.61,-35.92,46)

(-35.92,17.61,46)

Correct answer:

(14.98,37.09,46)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=40cos(68^{\circ})=14.98\\&y=40sin(68^{\circ})=37.09\\&z=46\end{align*}$

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Example Question #1862 : Calculus 3

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (113,-150^{\circ},-15).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(79.02,80.78,-15)

(-56.5,-97.86,-15)

(80.78,79.02,-15)

(-97.86,-56.5,-15)

Correct answer:

(-97.86,-56.5,-15)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=113cos(-150^{\circ})=-97.86\\&y=113sin(-150^{\circ})=-56.5\\&z=-15\end{align*}$

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Example Question #1863 : Calculus 3

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (35,-125^{\circ},124).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-20.08,-28.67,124)

(-28.67,-20.08,124)

(27.57,21.56,124)

(21.56,27.57,124)

Correct answer:

(-20.08,-28.67,124)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=35cos(-125^{\circ})=-20.08\\&y=35sin(-125^{\circ})=-28.67\\&z=124\end{align*}$

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Example Question #1864 : Calculus 3

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (124,179^{\circ},12).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(2.16,-123.98,12)

(-123.98,2.16,12)

(-123.69,8.77,12)

(8.77,-123.69,12)

Correct answer:

(-123.98,2.16,12)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=124cos(179^{\circ})=-123.98\\&y=124sin(179^{\circ})=2.16\\&z=12\end{align*}$

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Example Question #1865 : Calculus 3

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (12,-142^{\circ},-17).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-7.39,-9.46,-17)

(7.05,-9.71,-17)

(-9.71,7.05,-17)

(-9.46,-7.39,-17)

Correct answer:

(-9.46,-7.39,-17)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=12cos(-142^{\circ})=-9.46\\&y=12sin(-142^{\circ})=-7.39\\&z=-17\end{align*}$

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Example Question #1866 : Calculus 3

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (145,99^{\circ},-149).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(143.21,-22.68,-149)

(-144.88,5.77,-149)

(5.77,-144.88,-149)

(-22.68,143.21,-149)

Correct answer:

(-22.68,143.21,-149)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=145cos(99^{\circ})=-22.68\\&y=145sin(99^{\circ})=143.21\\&z=-149\end{align*}$

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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 #1855 : Calculus 3

Example Question #1856 : Calculus 3

Example Question #1857 : Calculus 3

Example Question #1858 : Calculus 3

Example Question #1861 : Calculus 3

Example Question #1862 : Calculus 3

Example Question #1863 : Calculus 3

Example Question #1864 : Calculus 3

Example Question #1865 : Calculus 3

Example Question #1866 : Calculus 3

All Calculus 3 Resources

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