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Calculus 3 : 3-Dimensional Space

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Example Questions

Example Question #201 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (60,108^{\circ},-72).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-18.54,57.06,-72)

(22.53,55.61,-72)

(55.61,22.53,-72)

(57.06,-18.54,-72)

Correct answer:

(-18.54,57.06,-72)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=60cos(108^{\circ})=-18.54\\&y=60sin(108^{\circ})=57.06\\&z=-72\end{align*}$

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Example Question #201 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (111,24,15).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(135,12.2^{\circ},15)$

$(113.56,167.8^{\circ},15)$

$(135,167.8^{\circ},15)$

$(113.56,12.2^{\circ},15)$

Correct answer:

$(113.56,12.2^{\circ},15)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms are the most straightforward }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{However, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\text{For our coordinates: }(111,24,15)\\&r=\sqrt{(111)^2+(24)^2}=113.56\\&\theta=arctan(\frac{24}{111})=12.2^{\circ}\\&z=15\end{align*}$

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Example Question #114 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-107,106,37).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(150.62,44.73^{\circ},37)$

$(1,44.73^{\circ},37)$

$(150.62,135.27^{\circ},37)$

$(1,135.27^{\circ},37)$

Correct answer:

$(150.62,135.27^{\circ},37)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-107,106,37)\\&r=\sqrt{(-107)^2+(106)^2}=150.62\\&\theta=arctan(\frac{106}{-107})=135.27^{\circ}\\&z=37\end{align*}$

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Example Question #115 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-45,4,-30).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(45.18,174.92^{\circ},-30)$

$(41,174.92^{\circ},-30)$

$(41,5.08^{\circ},-30)$

$(45.18,5.08^{\circ},-30)$

Correct answer:

$(45.18,174.92^{\circ},-30)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-45,4,-30)\\&r=\sqrt{(-45)^2+(4)^2}=45.18\\&\theta=arctan(\frac{4}{-45})=174.92^{\circ}\\&z=-30\end{align*}$

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Example Question #116 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-128,-78,-113).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(206,31.36^{\circ},-113)$

$(149.89,-148.64^{\circ},-113)$

$(206,-148.64^{\circ},-113)$

$(149.89,31.36^{\circ},-113)$

Correct answer:

$(149.89,-148.64^{\circ},-113)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-128,-78,-113)\\&r=\sqrt{(-128)^2+(-78)^2}=149.89\\&\theta=arctan(\frac{-78}{-128})=-148.64^{\circ}\\&z=-113\end{align*}$

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Example Question #117 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-95,-78,-25).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(122.92,-140.61^{\circ},-25)$

$(173,39.39^{\circ},-25)$

$(122.92,39.39^{\circ},-25)$

$(173,-140.61^{\circ},-25)$

Correct answer:

$(122.92,-140.61^{\circ},-25)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-95,-78,-25)\\&r=\sqrt{(-95)^2+(-78)^2}=122.92\\&\theta=arctan(\frac{-78}{-95})=-140.61^{\circ}\\&z=-25\end{align*}$

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Example Question #121 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-136,121,134).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(15,41.66^{\circ},134)$

$(182.04,41.66^{\circ},134)$

$(15,138.34^{\circ},134)$

$(182.04,138.34^{\circ},134)$

Correct answer:

$(182.04,138.34^{\circ},134)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-136,121,134)\\&r=\sqrt{(-136)^2+(121)^2}=182.04\\&\theta=arctan(\frac{121}{-136})=138.34^{\circ}\\&z=134\end{align*}$

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Example Question #202 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-3,-3,-49).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(4.24,-135^{\circ},-49)$

$(6,45^{\circ},-49)$

$(4.24,45^{\circ},-49)$

$(6,-135^{\circ},-49)$

Correct answer:

$(4.24,-135^{\circ},-49)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-3,-3,-49)\\&r=\sqrt{(-3)^2+(-3)^2}=4.24\\&\theta=arctan(\frac{-3}{-3})=-135^{\circ}\\&z=-49\end{align*}$

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Example Question #121 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (120,-39,-117).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(126.18,-18^{\circ},-117)$

$(81,-18^{\circ},-117)$

$(126.18,162^{\circ},-117)$

$(81,162^{\circ},-117)$

Correct answer:

$(126.18,-18^{\circ},-117)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(120,-39,-117)\\&r=\sqrt{(120)^2+(-39)^2}=126.18\\&\theta=arctan(\frac{-39}{120})=-18^{\circ}\\&z=-117\end{align*}$

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Example Question #122 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (84,-33,-78).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(90.25,-21.45^{\circ},-78)$

$(51,-21.45^{\circ},-78)$

$(51,158.55^{\circ},-78)$

$(90.25,158.55^{\circ},-78)$

Correct answer:

$(90.25,-21.45^{\circ},-78)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(84,-33,-78)\\&r=\sqrt{(84)^2+(-33)^2}=90.25\\&\theta=arctan(\frac{-33}{84})=-21.45^{\circ}\\&z=-78\end{align*}$

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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 #201 : 3 Dimensional Space

Example Question #201 : 3 Dimensional Space

Example Question #114 : Cylindrical Coordinates

Example Question #115 : Cylindrical Coordinates

Example Question #116 : Cylindrical Coordinates

Example Question #117 : Cylindrical Coordinates

Example Question #121 : Cylindrical Coordinates

Example Question #202 : 3 Dimensional Space

Example Question #121 : Cylindrical Coordinates

Example Question #122 : Cylindrical Coordinates

All Calculus 3 Resources

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