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Calculus 3 : 3-Dimensional Space

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Example Questions

Example Question #93 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (39,72^{\circ},2).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(12.05,37.09,2)

(37.09,12.05,2)

(-37.72,9.9,2)

(9.9,-37.72,2)

Correct answer:

(12.05,37.09,2)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=39cos(72^{\circ})=12.05\\&y=39sin(72^{\circ})=37.09\\&z=2\end{align*}$

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Example Question #94 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (134,17^{\circ},138).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(39.18,128.14,138)

(-36.87,-128.83,138)

(128.14,39.18,138)

(-128.83,-36.87,138)

Correct answer:

(128.14,39.18,138)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=134cos(17^{\circ})=128.14\\&y=134sin(17^{\circ})=39.18\\&z=138\end{align*}$

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Example Question #95 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (21,-88^{\circ},-106).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-0.74,20.99,-106)

(20.99,-0.74,-106)

(0.73,-20.99,-106)

(-20.99,0.73,-106)

Correct answer:

(0.73,-20.99,-106)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=21cos(-88^{\circ})=0.73\\&y=21sin(-88^{\circ})=-20.99\\&z=-106\end{align*}$

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Example Question #96 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (127,113^{\circ},-74).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(126.4,-12.34,-74)

(116.9,-49.62,-74)

(-12.34,126.4,-74)

(-49.62,116.9,-74)

Correct answer:

(-49.62,116.9,-74)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=127cos(113^{\circ})=-49.62\\&y=127sin(113^{\circ})=116.9\\&z=-74\end{align*}$

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Example Question #97 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (37,-54^{\circ},129).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(20.68,-30.68,129)

(21.75,-29.93,129)

(-30.68,20.68,129)

(-29.93,21.75,129)

Correct answer:

(21.75,-29.93,129)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=37cos(-54^{\circ})=21.75\\&y=37sin(-54^{\circ})=-29.93\\&z=129\end{align*}$

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Example Question #98 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (30,42^{\circ},-75).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(20.07,22.29,-75)

(-12,-27.5,-75)

(-27.5,-12,-75)

(22.29,20.07,-75)

Correct answer:

(22.29,20.07,-75)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=30cos(42^{\circ})=22.29\\&y=30sin(42^{\circ})=20.07\\&z=-75\end{align*}$

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Example Question #1851 : Calculus 3

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (71,119^{\circ},-45).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(62.1,-34.42,-45)

(-26.37,65.92,-45)

(65.92,-26.37,-45)

(-34.42,62.1,-45)

Correct answer:

(-34.42,62.1,-45)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=71cos(119^{\circ})=-34.42\\&y=71sin(119^{\circ})=62.1\\&z=-45\end{align*}$

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Example Question #1852 : Calculus 3

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (88,151^{\circ},15).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(42.66,-76.97,15)

(-76.97,42.66,15)

(86.18,17.79,15)

(17.79,86.18,15)

Correct answer:

(-76.97,42.66,15)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=88cos(151^{\circ})=-76.97\\&y=88sin(151^{\circ})=42.66\\&z=15\end{align*}$

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Example Question #1853 : Calculus 3

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (43,92^{\circ},77).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-33.52,-26.94,77)

(42.97,-1.5,77)

(-1.5,42.97,77)

(-26.94,-33.52,77)

Correct answer:

(-1.5,42.97,77)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=43cos(92^{\circ})=-1.5\\&y=43sin(92^{\circ})=42.97\\&z=77\end{align*}$

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Example Question #1854 : Calculus 3

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (58,-153^{\circ},20).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-51.68,-26.33,20)

(-34.3,-46.77,20)

(-26.33,-51.68,20)

(-46.77,-34.3,20)

Correct answer:

(-51.68,-26.33,20)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=58cos(-153^{\circ})=-51.68\\&y=58sin(-153^{\circ})=-26.33\\&z=20\end{align*}$

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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 #93 : Cylindrical Coordinates

Example Question #94 : Cylindrical Coordinates

Example Question #95 : Cylindrical Coordinates

Example Question #96 : Cylindrical Coordinates

Example Question #97 : Cylindrical Coordinates

Example Question #98 : Cylindrical Coordinates

Example Question #1851 : Calculus 3

Example Question #1852 : Calculus 3

Example Question #1853 : Calculus 3

Example Question #1854 : Calculus 3

All Calculus 3 Resources

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