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Calculus 3 : 3-Dimensional Space

Study concepts, example questions & explanations for Calculus 3

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6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #171 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (45,30,-29) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(54.08,33.69^{\circ},-29)$

$(54.08,146.31^{\circ},-29)$

$(75,33.69^{\circ},-29)$

$(75,146.31^{\circ},-29)$

Correct answer:

$(54.08,33.69^{\circ},-29)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

It is something to bear in mind when making a calculation using a calculator; negative values by convention create a negative $\theta$ , while negative values lead to $|\theta|>90^{\circ};\frac{\pi}{2}$

For our coordinates (45,30,-29)

$\begin{align*}&r=\sqrt{(45)^2+(30)^2}=54.08\\&\theta=arctan(\frac{30}{45})=33.69^{\circ}\\&z=-29\end{align*}$

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Example Question #172 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (48,17,29) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(65,160.5^{\circ},29)$

$(50.92,160.5^{\circ},29)$

$(50.92,19.5^{\circ},29)$

Correct answer:

$(50.92,19.5^{\circ},29)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (48,17,29)

$\begin{align*}&r=\sqrt{(48)^2+(17)^2}=50.92\\&\theta=arctan(\frac{17}{48})=19.5^{\circ}\\&z=29\end{align*}$

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Example Question #173 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (-12,7,-14) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(5,149.74^{\circ},-14)$

$(13.89,149.74^{\circ},-14)$

$(13.89,-30.26^{\circ},-14)$

$(5,-30.26^{\circ},-14)$

Correct answer:

$(13.89,149.74^{\circ},-14)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (-12,7,-14)

$\begin{align*}&r=\sqrt{(-12)^2+(7)^2}=13.89\\&\theta=arctan(\frac{7}{-12})=149.74^{\circ}\\&z=-14\end{align*}$

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Example Question #174 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (17,21,15) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(38,128.99^{\circ},15)$

$(38,51.01^{\circ},15)$

$(27.02,128.99^{\circ},15)$

$(27.02,51.01^{\circ},15)$

Correct answer:

$(27.02,51.01^{\circ},15)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (17,21,15)

$\begin{align*}&r=\sqrt{(17)^2+(21)^2}=27.02\\&\theta=arctan(\frac{21}{17})=51.01^{\circ}\\&z=15\end{align*}$

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Example Question #175 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (50,-47,-27) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(3,136.77^{\circ},-27)$

$(68.62,136.77^{\circ},-27)$

$(3,-43.23^{\circ},-27)$

$(68.62,-43.23^{\circ},-27)$

Correct answer:

$(68.62,-43.23^{\circ},-27)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (50,-47,-27)

$\begin{align*}&r=\sqrt{(50)^2+(-47)^2}=68.62\\&\theta=arctan(\frac{-47}{50})=-43.23^{\circ}\\&z=-27\end{align*}$

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Example Question #176 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (48,7,-19) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(48.51,171.7^{\circ},-19)$

$(48.51,8.3^{\circ},-19)$

$(55,8.3^{\circ},-19)$

$(55,171.7^{\circ},-19)$

Correct answer:

$(48.51,8.3^{\circ},-19)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (48,7,-19)

$\begin{align*}&r=\sqrt{(48)^2+(7)^2}=48.51\\&\theta=arctan(\frac{7}{48})=8.3^{\circ}\\&z=-19\end{align*}$

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Example Question #1841 : Calculus 3

A point in space is located, in Cartesian coordinates, at (2,-6,13) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(6.32,-71.57^{\circ},13)$

$(4,-71.57^{\circ},13)$

$(4,108.43^{\circ},13)$

$(6.32,108.43^{\circ},13)$

Correct answer:

$(6.32,-71.57^{\circ},13)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (2,-6,13)

$\begin{align*}&r=\sqrt{(2)^2+(-6)^2}=6.32\\&\theta=arctan(\frac{-6}{2})=-71.57^{\circ}\\&z=13\end{align*}$

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Example Question #1842 : Calculus 3

A point in space is located, in Cartesian coordinates, at (-2,-39,30) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(41,-92.94^{\circ},30)$

$(39.05,87.06^{\circ},30)$

$(39.05,-92.94^{\circ},30)$

$(41,87.06^{\circ},30)$

Correct answer:

$(39.05,-92.94^{\circ},30)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (-2,-39,30)

$\begin{align*}&r=\sqrt{(-2)^2+(-39)^2}=39.05\\&\theta=arctan(\frac{-39}{-2})=-92.94^{\circ}\\&z=30\end{align*}$

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Example Question #177 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (75,-58^{\circ},138).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-63.6,39.74,138)

(39.74,-63.6,138)

(-74.47,8.94,138)

(8.94,-74.47,138)

Correct answer:

(39.74,-63.6,138)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=75cos(-58^{\circ})=39.74\\&y=75sin(-58^{\circ})=-63.6\\&z=138\end{align*}$

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Example Question #178 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (88,91^{\circ},-83).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(87.99,-1.54,-83)

(-87.5,9.33,-83)

(-1.54,87.99,-83)

(9.33,-87.5,-83)

Correct answer:

(-1.54,87.99,-83)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=88cos(91^{\circ})=-1.54\\&y=88sin(91^{\circ})=87.99\\&z=-83\end{align*}$

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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 #171 : 3 Dimensional Space

Example Question #172 : 3 Dimensional Space

Example Question #173 : 3 Dimensional Space

Example Question #174 : 3 Dimensional Space

Example Question #175 : 3 Dimensional Space

Example Question #176 : 3 Dimensional Space

Example Question #1841 : Calculus 3

Example Question #1842 : Calculus 3

Example Question #177 : 3 Dimensional Space

Example Question #178 : 3 Dimensional Space

All Calculus 3 Resources

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