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Calculus 3 : 3-Dimensional Space

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Example Questions

Example Question #281 : 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:

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

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

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

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

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 #282 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-140,105,131).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(35,-36.87^{\circ},131)$

$(175,-36.87^{\circ},131)$

$(35,143.13^{\circ},131)$

$(175,143.13^{\circ},131)$

Correct answer:

$(175,143.13^{\circ},131)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-140,105,131)\\&r=\sqrt{(-140)^2+(105)^2}=175\\&\theta=arctan(\frac{105}{-140})=143.13^{\circ}\\&z=131\end{align*}$

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Example Question #283 : 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:

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

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

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

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

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 #284 : 3 Dimensional Space

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

Possible Answers:

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

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

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

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

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 #285 : 3 Dimensional Space

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

Possible Answers:

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

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

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

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

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 #286 : 3 Dimensional Space

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

Possible Answers:

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

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

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

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

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 #287 : 3 Dimensional Space

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

Possible Answers:

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

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

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

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

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 #288 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-140,-18,-36).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(158,7.33^{\circ},-36)$

$(141.15,-172.67^{\circ},-36)$

$(141.15,7.33^{\circ},-36)$

$(158,-172.67^{\circ},-36)$

Correct answer:

$(141.15,-172.67^{\circ},-36)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-140,-18,-36)\\&r=\sqrt{(-140)^2+(-18)^2}=141.15\\&\theta=arctan(\frac{-18}{-140})=-172.67^{\circ}\\&z=-36\end{align*}$

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Example Question #289 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (80,89,-94).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(119.67,-131.95^{\circ},-94)$

$(169,-131.95^{\circ},-94)$

$(169,48.05^{\circ},-94)$

$(119.67,48.05^{\circ},-94)$

Correct answer:

$(119.67,48.05^{\circ},-94)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(80,89,-94)\\&r=\sqrt{(80)^2+(89)^2}=119.67\\&\theta=arctan(\frac{89}{80})=48.05^{\circ}\\&z=-94\end{align*}$

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Example Question #290 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-3,-16,44).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(16.28,79.38^{\circ},44)$

$(19,79.38^{\circ},44)$

$(19,-100.62^{\circ},44)$

$(16.28,-100.62^{\circ},44)$

Correct answer:

$(16.28,-100.62^{\circ},44)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-3,-16,44)\\&r=\sqrt{(-3)^2+(-16)^2}=16.28\\&\theta=arctan(\frac{-16}{-3})=-100.62^{\circ}\\&z=44\end{align*}$

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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 #281 : 3 Dimensional Space

Example Question #282 : 3 Dimensional Space

Example Question #283 : 3 Dimensional Space

Example Question #284 : 3 Dimensional Space

Example Question #285 : 3 Dimensional Space

Example Question #286 : 3 Dimensional Space

Example Question #287 : 3 Dimensional Space

Example Question #288 : 3 Dimensional Space

Example Question #289 : 3 Dimensional Space

Example Question #290 : 3 Dimensional Space

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