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Calculus 3 : 3-Dimensional Space

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #235 : Cylindrical Coordinates

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

Possible Answers:

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

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

$(154.03,156.26^{\circ},-67)$

$(154.03,-23.74^{\circ},-67)$

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 #236 : Cylindrical Coordinates

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

Possible Answers:

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

$(182.78,-131.45^{\circ},97)$

$(182.78,48.55^{\circ},97)$

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

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 #237 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (105,163^{\circ},-55).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-100.41,30.7,-55)

(98.16,-37.27,-55)

(-37.27,98.16,-55)

(30.7,-100.41,-55)

Correct answer:

(-100.41,30.7,-55)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=105cos(163^{\circ})=-100.41\\&y=105sin(163^{\circ})=30.7\\&z=-55\end{align*}$

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Example Question #238 : 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:

(4.39,-4.09,-18)

(4.99,3.33,-18)

(-4.09,4.39,-18)

(3.33,4.99,-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 #1991 : 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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Example Question #1992 : Calculus 3

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (74,53^{\circ},-16).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(59.1,44.53,-16)

(-67.95,29.3,-16)

(29.3,-67.95,-16)

(44.53,59.1,-16)

Correct answer:

(44.53,59.1,-16)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=74cos(53^{\circ})=44.53\\&y=74sin(53^{\circ})=59.1\\&z=-16\end{align*}$

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Example Question #241 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (107,-81^{\circ},77).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(67.4,83.11,77)

(-105.68,16.74,77)

(16.74,-105.68,77)

(83.11,67.4,77)

Correct answer:

(16.74,-105.68,77)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=107cos(-81^{\circ})=16.74\\&y=107sin(-81^{\circ})=-105.68\\&z=77\end{align*}$

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Example Question #242 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (102,-122^{\circ},47).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-50.87,-88.41,47)

(-86.5,-54.05,47)

(-54.05,-86.5,47)

(-88.41,-50.87,47)

Correct answer:

(-54.05,-86.5,47)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=102cos(-122^{\circ})=-54.05\\&y=102sin(-122^{\circ})=-86.5\\&z=47\end{align*}$

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Example Question #243 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (18,166^{\circ},0).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-15.76,8.7,0)

(-17.47,4.35,0)

(8.7,-15.76,0)

(4.35,-17.47,0)

Correct answer:

(-17.47,4.35,0)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=18cos(166^{\circ})=-17.47\\&y=18sin(166^{\circ})=4.35\\&z=0\end{align*}$

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Example Question #244 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (52,-100^{\circ},26).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-51.21,-9.03,26)

(26.33,44.84,26)

(44.84,26.33,26)

(-9.03,-51.21,26)

Correct answer:

(-9.03,-51.21,26)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=52cos(-100^{\circ})=-9.03\\&y=52sin(-100^{\circ})=-51.21\\&z=26\end{align*}$

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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 #235 : Cylindrical Coordinates

Example Question #236 : Cylindrical Coordinates

Example Question #237 : Cylindrical Coordinates

Example Question #238 : Cylindrical Coordinates

Example Question #1991 : Calculus 3

Example Question #1992 : Calculus 3

Example Question #241 : Cylindrical Coordinates

Example Question #242 : Cylindrical Coordinates

Example Question #243 : Cylindrical Coordinates

Example Question #244 : Cylindrical Coordinates

All Calculus 3 Resources

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