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Calculus 3 : 3-Dimensional Space

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Example Questions

Example Question #211 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-29,-121,-111).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(150,76.52^{\circ},-111)$

$(124.43,-103.48^{\circ},-111)$

$(150,-103.48^{\circ},-111)$

$(124.43,76.52^{\circ},-111)$

Correct answer:

$(124.43,-103.48^{\circ},-111)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-29,-121,-111)\\&r=\sqrt{(-29)^2+(-121)^2}=124.43\\&\theta=arctan(\frac{-121}{-29})=-103.48^{\circ}\\&z=-111\end{align*}$

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Example Question #212 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (133,137,23).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(270,134.15^{\circ},23)$

$(190.94,134.15^{\circ},23)$

$(190.94,45.85^{\circ},23)$

$(270,45.85^{\circ},23)$

Correct answer:

$(190.94,45.85^{\circ},23)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(133,137,23)\\&r=\sqrt{(133)^2+(137)^2}=190.94\\&\theta=arctan(\frac{137}{133})=45.85^{\circ}\\&z=23\end{align*}$

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Example Question #121 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-133,-80,-44).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(213,31.03^{\circ},-44)$

$(213,-148.97^{\circ},-44)$

$(155.21,-148.97^{\circ},-44)$

$(155.21,31.03^{\circ},-44)$

Correct answer:

$(155.21,-148.97^{\circ},-44)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-133,-80,-44)\\&r=\sqrt{(-133)^2+(-80)^2}=155.21\\&\theta=arctan(\frac{-80}{-133})=-148.97^{\circ}\\&z=-44\end{align*}$

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Example Question #214 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-37,-93,-22).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(100.09,68.3^{\circ},-22)$

$(100.09,-111.7^{\circ},-22)$

$(130,68.3^{\circ},-22)$

$(130,-111.7^{\circ},-22)$

Correct answer:

$(100.09,-111.7^{\circ},-22)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-37,-93,-22)\\&r=\sqrt{(-37)^2+(-93)^2}=100.09\\&\theta=arctan(\frac{-93}{-37})=-111.7^{\circ}\\&z=-22\end{align*}$

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Example Question #215 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-5,-114,27).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(119,87.49^{\circ},27)$

$(114.11,-92.51^{\circ},27)$

$(114.11,87.49^{\circ},27)$

$(119,-92.51^{\circ},27)$

Correct answer:

$(114.11,-92.51^{\circ},27)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-5,-114,27)\\&r=\sqrt{(-5)^2+(-114)^2}=114.11\\&\theta=arctan(\frac{-114}{-5})=-92.51^{\circ}\\&z=27\end{align*}$

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Example Question #216 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-82,-35,25).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(89.16,23.11^{\circ},25)$

$(89.16,-156.89^{\circ},25)$

$(117,-156.89^{\circ},25)$

$(117,23.11^{\circ},25)$

Correct answer:

$(89.16,-156.89^{\circ},25)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-82,-35,25)\\&r=\sqrt{(-82)^2+(-35)^2}=89.16\\&\theta=arctan(\frac{-35}{-82})=-156.89^{\circ}\\&z=25\end{align*}$

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Example Question #131 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-75,-63,35).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(97.95,40.03^{\circ},35)$

$(97.95,-139.97^{\circ},35)$

$(138,40.03^{\circ},35)$

$(138,-139.97^{\circ},35)$

Correct answer:

$(97.95,-139.97^{\circ},35)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-75,-63,35)\\&r=\sqrt{(-75)^2+(-63)^2}=97.95\\&\theta=arctan(\frac{-63}{-75})=-139.97^{\circ}\\&z=35\end{align*}$

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Example Question #131 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-71,98,145).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(121.02,125.92^{\circ},145)$

$(27,54.08^{\circ},145)$

$(27,125.92^{\circ},145)$

$(121.02,54.08^{\circ},145)$

Correct answer:

$(121.02,125.92^{\circ},145)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-71,98,145)\\&r=\sqrt{(-71)^2+(98)^2}=121.02\\&\theta=arctan(\frac{98}{-71})=125.92^{\circ}\\&z=145\end{align*}$

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Example Question #132 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (69,-47,25).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(22,-34.26^{\circ},25)$

$(83.49,-34.26^{\circ},25)$

$(83.49,145.74^{\circ},25)$

$(22,145.74^{\circ},25)$

Correct answer:

$(83.49,-34.26^{\circ},25)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(69,-47,25)\\&r=\sqrt{(69)^2+(-47)^2}=83.49\\&\theta=arctan(\frac{-47}{69})=-34.26^{\circ}\\&z=25\end{align*}$

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Example Question #132 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (17,137^{\circ},122).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-12.43,11.59,122)

(-16.02,5.68,122)

(5.68,-16.02,122)

(11.59,-12.43,122)

Correct answer:

(-12.43,11.59,122)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=17cos(137^{\circ})=-12.43\\&y=17sin(137^{\circ})=11.59\\&z=122\end{align*}$

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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 #211 : 3 Dimensional Space

Example Question #212 : 3 Dimensional Space

Example Question #121 : Cylindrical Coordinates

Example Question #214 : 3 Dimensional Space

Example Question #215 : 3 Dimensional Space

Example Question #216 : 3 Dimensional Space

Example Question #131 : Cylindrical Coordinates

Example Question #131 : Cylindrical Coordinates

Example Question #132 : Cylindrical Coordinates

Example Question #132 : Cylindrical Coordinates

All Calculus 3 Resources

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