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Calculus 3 : 3-Dimensional Space

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Example Questions

Example Question #261 : 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)

(54.66,-75.24,-8)

(-77.13,51.97,-8)

(-75.24,54.66,-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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Example Question #262 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (125,18^{\circ},26).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(38.63,118.88,26)

(-93.87,82.54,26)

(118.88,38.63,26)

(82.54,-93.87,26)

Correct answer:

(118.88,38.63,26)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=125cos(18^{\circ})=118.88\\&y=125sin(18^{\circ})=38.63\\&z=26\end{align*}$

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Example Question #263 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (76,-36,20).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(40,-25.35^{\circ},20)$

$(84.1,154.65^{\circ},20)$

$(84.1,-25.35^{\circ},20)$

$(40,154.65^{\circ},20)$

Correct answer:

$(84.1,-25.35^{\circ},20)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(76,-36,20)\\&r=\sqrt{(76)^2+(-36)^2}=84.1\\&\theta=arctan(\frac{-36}{76})=-25.35^{\circ}\\&z=20\end{align*}$

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Example Question #264 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-128,-134,9).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(185.31,-133.69^{\circ},9)$

$(262,46.31^{\circ},9)$

$(185.31,46.31^{\circ},9)$ $(262,-133.69^{\circ},9)$

$(262,-133.69^{\circ},9)$

Correct answer:

$(185.31,-133.69^{\circ},9)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-128,-134,9)\\&r=\sqrt{(-128)^2+(-134)^2}=185.31\\&\theta=arctan(\frac{-134}{-128})=-133.69^{\circ}\\&z=9\end{align*}$

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Example Question #265 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (84,131,-111).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(155.62,57.33^{\circ},-111)$

$(215,57.33^{\circ},-111)$

$(155.62,122.67^{\circ},-111)$

$(215,122.67^{\circ},-111)$

Correct answer:

$(155.62,57.33^{\circ},-111)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(84,131,-111)\\&r=\sqrt{(84)^2+(131)^2}=155.62\\&\theta=arctan(\frac{131}{84})=57.33^{\circ}\\&z=-111\end{align*}$

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Example Question #266 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (21,-9,-147).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(22.85,-23.2^{\circ},-147)$

$(12,-23.2^{\circ},-147)$

$(12,156.8^{\circ},-147)$

$(22.85,156.8^{\circ},-147)$

Correct answer:

$(22.85,-23.2^{\circ},-147)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(21,-9,-147)\\&r=\sqrt{(21)^2+(-9)^2}=22.85\\&\theta=arctan(\frac{-9}{21})=-23.2^{\circ}\\&z=-147\end{align*}$

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Example Question #267 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-49,-102,89).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(151,-115.66^{\circ},89)$

$(113.16,-115.66^{\circ},89)$

$(151,64.34^{\circ},89)$

$(113.16,64.34^{\circ},89)$

Correct answer:

$(113.16,-115.66^{\circ},89)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-49,-102,89)\\&r=\sqrt{(-49)^2+(-102)^2}=113.16\\&\theta=arctan(\frac{-102}{-49})=-115.66^{\circ}\\&z=89\end{align*}$

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Example Question #181 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-57,9,-101).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(57.71,171.03^{\circ},-101)$

$(48,8.97^{\circ},-101)$

$(48,171.03^{\circ},-101)$

$(57.71,8.97^{\circ},-101)$

Correct answer:

$(57.71,171.03^{\circ},-101)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-57,9,-101)\\&r=\sqrt{(-57)^2+(9)^2}=57.71\\&\theta=arctan(\frac{9}{-57})=171.03^{\circ}\\&z=-101\end{align*}$

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Example Question #268 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (31,-71,46).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(40,113.59^{\circ},46)$

$(77.47,-66.41^{\circ},46)$

$(40,-66.41^{\circ},46)$

$(77.47,113.59^{\circ},46)$

Correct answer:

$(77.47,-66.41^{\circ},46)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(31,-71,46)\\&r=\sqrt{(31)^2+(-71)^2}=77.47\\&\theta=arctan(\frac{-71}{31})=-66.41^{\circ}\\&z=46\end{align*}$

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Example Question #182 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (57,75,-15).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(94.2,52.77^{\circ},-15)$

$(132,127.23^{\circ},-15)$

$(94.2,127.23^{\circ},-15)$

$(132,52.77^{\circ},-15)$

Correct answer:

$(94.2,52.77^{\circ},-15)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(57,75,-15)\\&r=\sqrt{(57)^2+(75)^2}=94.2\\&\theta=arctan(\frac{75}{57})=52.77^{\circ}\\&z=-15\end{align*}$

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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 #261 : 3 Dimensional Space

Example Question #262 : 3 Dimensional Space

Example Question #263 : 3 Dimensional Space

Example Question #264 : 3 Dimensional Space

Example Question #265 : 3 Dimensional Space

Example Question #266 : 3 Dimensional Space

Example Question #267 : 3 Dimensional Space

Example Question #181 : Cylindrical Coordinates

Example Question #268 : 3 Dimensional Space

Example Question #182 : Cylindrical Coordinates

All Calculus 3 Resources

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