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Calculus 3 : 3-Dimensional Space

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Example Questions

Example Question #231 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (138,-4,90).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(138.06,178.34^{\circ},90)$

$(134,-1.66^{\circ},90)$

$(138.06,-1.66^{\circ},90)$

$(134,178.34^{\circ},90)$

Correct answer:

$(138.06,-1.66^{\circ},90)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(138,-4,90)\\&r=\sqrt{(138)^2+(-4)^2}=138.06\\&\theta=arctan(\frac{-4}{138})=-1.66^{\circ}\\&z=90\end{align*}$

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Example Question #232 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-108,-24,125).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(132,12.53^{\circ},125)$

$(132,-167.47^{\circ},125)$

$(110.63,12.53^{\circ},125)$

$(110.63,-167.47^{\circ},125)$

Correct answer:

$(110.63,-167.47^{\circ},125)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-108,-24,125)\\&r=\sqrt{(-108)^2+(-24)^2}=110.63\\&\theta=arctan(\frac{-24}{-108})=-167.47^{\circ}\\&z=125\end{align*}$

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Example Question #233 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (88,138,47).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(163.67,57.48^{\circ},47)$

$(226,122.52^{\circ},47)$

$(226,57.48^{\circ},47)$

$(163.67,122.52^{\circ},47)$

Correct answer:

$(163.67,57.48^{\circ},47)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(88,138,47)\\&r=\sqrt{(88)^2+(138)^2}=163.67\\&\theta=arctan(\frac{138}{88})=57.48^{\circ}\\&z=47\end{align*}$

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Example Question #234 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (54,78,73).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(94.87,55.3^{\circ},73)$

$(132,124.7^{\circ},73)$

$(94.87,124.7^{\circ},73)$

$(132,55.3^{\circ},73)$

Correct answer:

$(94.87,55.3^{\circ},73)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(54,78,73)\\&r=\sqrt{(54)^2+(78)^2}=94.87\\&\theta=arctan(\frac{78}{54})=55.3^{\circ}\\&z=73\end{align*}$

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Example Question #143 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-32,47,-99).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(15,55.75^{\circ},-99)$

$(56.86,55.75^{\circ},-99)$

$(15,124.25^{\circ},-99)$

$(56.86,124.25^{\circ},-99)$

Correct answer:

$(56.86,124.25^{\circ},-99)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-32,47,-99)\\&r=\sqrt{(-32)^2+(47)^2}=56.86\\&\theta=arctan(\frac{47}{-32})=124.25^{\circ}\\&z=-99\end{align*}$

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Example Question #1901 : Calculus 3

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (62,-141,-67).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(79,-66.26^{\circ},-67)$

$(154.03,113.74^{\circ},-67)$

$(79,113.74^{\circ},-67)$

$(154.03,-66.26^{\circ},-67)$

Correct answer:

$(154.03,-66.26^{\circ},-67)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(62,-141,-67)\\&r=\sqrt{(62)^2+(-141)^2}=154.03\\&\theta=arctan(\frac{-141}{62})=-66.26^{\circ}\\&z=-67\end{align*}$

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Example Question #1902 : Calculus 3

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-137,-121,97).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(258,-138.55^{\circ},97)$

$(182.78,41.45^{\circ},97)$

$(258,41.45^{\circ},97)$

$(182.78,-138.55^{\circ},97)$

Correct answer:

$(182.78,-138.55^{\circ},97)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-137,-121,97)\\&r=\sqrt{(-137)^2+(-121)^2}=182.78\\&\theta=arctan(\frac{-121}{-137})=-138.55^{\circ}\\&z=97\end{align*}$

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Example Question #1903 : Calculus 3

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (59,-55,136).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(4,-42.99^{\circ},136)$

$(4,137.01^{\circ},136)$

$(80.66,137.01^{\circ},136)$

$(80.66,-42.99^{\circ},136)$

Correct answer:

$(80.66,-42.99^{\circ},136)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(59,-55,136)\\&r=\sqrt{(59)^2+(-55)^2}=80.66\\&\theta=arctan(\frac{-55}{59})=-42.99^{\circ}\\&z=136\end{align*}$

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Example Question #151 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (6,-43^{\circ},-18).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(4.99,3.33,-18)

(-4.09,4.39,-18)

(3.33,4.99,-18)

(4.39,-4.09,-18)

Correct answer:

(4.39,-4.09,-18)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=6cos(-43^{\circ})=4.39\\&y=6sin(-43^{\circ})=-4.09\\&z=-18\end{align*}$

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Example Question #1904 : Calculus 3

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (115,-113^{\circ},89).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(11.18,114.46,89)

(-44.93,-105.86,89)

(-105.86,-44.93,89)

(114.46,11.18,89)

Correct answer:

(-44.93,-105.86,89)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=115cos(-113^{\circ})=-44.93\\&y=115sin(-113^{\circ})=-105.86\\&z=89\end{align*}$

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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 #231 : 3 Dimensional Space

Example Question #232 : 3 Dimensional Space

Example Question #233 : 3 Dimensional Space

Example Question #234 : 3 Dimensional Space

Example Question #143 : Cylindrical Coordinates

Example Question #1901 : Calculus 3

Example Question #1902 : Calculus 3

Example Question #1903 : Calculus 3

Example Question #151 : Cylindrical Coordinates

Example Question #1904 : Calculus 3

All Calculus 3 Resources

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