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Calculus 3 : 3-Dimensional Space

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Example Questions

Example Question #204 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (63,77,-67).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(99.49,50.71^{\circ},-67)$

$(140,50.71^{\circ},-67)$

$(99.49,-129.29^{\circ},-67)$

$(140,-129.29^{\circ},-67)$

Correct answer:

$(99.49,50.71^{\circ},-67)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(63,77,-67)\\&r=\sqrt{(63)^2+(77)^2}=99.49\\&\theta=arctan(\frac{77}{63})=50.71^{\circ}\\&z=-67\end{align*}$

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Example Question #205 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (54,47,-102).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(71.59,41.04^{\circ},-102)$

$(71.59,-138.96^{\circ},-102)$

$(101,-138.96^{\circ},-102)$

$(101,41.04^{\circ},-102)$

Correct answer:

$(71.59,41.04^{\circ},-102)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(54,47,-102)\\&r=\sqrt{(54)^2+(47)^2}=71.59\\&\theta=arctan(\frac{47}{54})=41.04^{\circ}\\&z=-102\end{align*}$

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Example Question #206 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-48,26,-83).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(22,151.56^{\circ},-83)$

$(54.59,-28.44^{\circ},-83)$

$(54.59,151.56^{\circ},-83)$

$(22,-28.44^{\circ},-83)$

Correct answer:

$(54.59,151.56^{\circ},-83)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-48,26,-83)\\&r=\sqrt{(-48)^2+(26)^2}=54.59\\&\theta=arctan(\frac{26}{-48})=151.56^{\circ}\\&z=-83\end{align*}$

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Example Question #207 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (76,-74,2).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(2,-44.24^{\circ},2)$

$(106.08,135.76^{\circ},2)$

$(106.08,-44.24^{\circ},2)$

$(2,135.76^{\circ},2)$

Correct answer:

$(106.08,-44.24^{\circ},2)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(76,-74,2)\\&r=\sqrt{(76)^2+(-74)^2}=106.08\\&\theta=arctan(\frac{-74}{76})=-44.24^{\circ}\\&z=2\end{align*}$

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Example Question #295 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (60,118,138).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(178,-116.95^{\circ},138)$

$(132.38,63.05^{\circ},138)$

$(178,63.05^{\circ},138)$

$(132.38,-116.95^{\circ},138)$

Correct answer:

$(132.38,63.05^{\circ},138)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(60,118,138)\\&r=\sqrt{(60)^2+(118)^2}=132.38\\&\theta=arctan(\frac{118}{60})=63.05^{\circ}\\&z=138\end{align*}$

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Example Question #296 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (14,-109,-106).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(95,-82.68^{\circ},-106)$

$(109.9,-82.68^{\circ},-106)$

$(109.9,97.32^{\circ},-106)$

$(95,97.32^{\circ},-106)$

Correct answer:

$(109.9,-82.68^{\circ},-106)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(14,-109,-106)\\&r=\sqrt{(14)^2+(-109)^2}=109.9\\&\theta=arctan(\frac{-109}{14})=-82.68^{\circ}\\&z=-106\end{align*}$

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Example Question #211 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-73,103,-74).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(30,125.33^{\circ},-74)$

$(30,-54.67^{\circ},-74)$

$(126.25,-54.67^{\circ},-74)$

$(126.25,125.33^{\circ},-74)$

Correct answer:

$(126.25,125.33^{\circ},-74)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-73,103,-74)\\&r=\sqrt{(-73)^2+(103)^2}=126.25\\&\theta=arctan(\frac{103}{-73})=125.33^{\circ}\\&z=-74\end{align*}$

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Example Question #212 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (95,-77,129).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(122.29,140.97^{\circ},129)$

$(122.29,-39.03^{\circ},129)$

$(18,-39.03^{\circ},129)$

$(18,140.97^{\circ},129)$

Correct answer:

$(122.29,-39.03^{\circ},129)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(95,-77,129)\\&r=\sqrt{(95)^2+(-77)^2}=122.29\\&\theta=arctan(\frac{-77}{95})=-39.03^{\circ}\\&z=129\end{align*}$

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Example Question #291 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (53,-90^{\circ},-91).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-47.38,-23.75,-91)

(0,-53,-91)

(-53,0,-91)

(-23.75,-47.38,-91)

Correct answer:

(0,-53,-91)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=53cos(-90^{\circ})=0\\&y=53sin(-90^{\circ})=-53\\&z=-91\end{align*}$

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Example Question #292 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (93,-54^{\circ},-8).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(51.97,-77.13,-8)

(-75.24,54.66,-8)

(54.66,-75.24,-8)

(-77.13,51.97,-8)

Correct answer:

(54.66,-75.24,-8)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=93cos(-54^{\circ})=54.66\\&y=93sin(-54^{\circ})=-75.24\\&z=-8\end{align*}$

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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 #204 : Cylindrical Coordinates

Example Question #205 : Cylindrical Coordinates

Example Question #206 : Cylindrical Coordinates

Example Question #207 : Cylindrical Coordinates

Example Question #295 : 3 Dimensional Space

Example Question #296 : 3 Dimensional Space

Example Question #211 : Cylindrical Coordinates

Example Question #212 : Cylindrical Coordinates

Example Question #291 : 3 Dimensional Space

Example Question #292 : 3 Dimensional Space

All Calculus 3 Resources

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