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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 #35 : Cylindrical Coordinates

A point in space is located, in cylindrical coordinates, at $(4,110^{\circ},6)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(-1.37,3.76,6)

(-0.177,-4,6)

(3.76,-1.37,6)

(1.37,-3.76,6)

(-4,-0.177,6)

Correct answer:

(-1.37,3.76,6)

Explanation:

When given cylindrical coordinates of the form $(r,\theta,z)$ and converting to Cartesian coordinates of the form (x,y,z) , the relationship between , , , and $\theta$ will be of use:

$x=rcos(\theta)$

$y=rsin(\theta)$

z=z (This value is identical across Cartesian and cylindrical coordinate systems)

For the cylindrical coordinates $(4,110^{\circ},6)$

When finding the Cartesian coordinates, be sure if using a calculator to have the correct angle units set (degrees vs radians):

$x=rcos(\theta)\rightarrow-1.37$

$y=rsin(\theta)\rightarrow3.76$

$z=z\rightarrow6$

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Example Question #36 : Cylindrical Coordinates

A point in space is located, in cylindrical coordinates, at $(8,150^{\circ},8)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(-5.72,5.59,8)

(4,6.93,8)

(4,-6.93,8)

(5.59,-5.72,8)

(-6.93,4,8)

Correct answer:

(-6.93,4,8)

Explanation:

When given cylindrical coordinates of the form $(r,\theta,z)$ and converting to Cartesian coordinates of the form (x,y,z) , the relationship between , , , and $\theta$ will be of use:

$x=rcos(\theta)$

$y=rsin(\theta)$

z=z (This value is identical across Cartesian and cylindrical coordinate systems)

For the cylindrical coordinates $(8,150^{\circ},8)$

When finding the Cartesian coordinates, be sure if using a calculator to have the correct angle units set (degrees vs radians):

$x=rcos(\theta)\rightarrow-6.93$

$y=rsin(\theta)\rightarrow4$

$z=z\rightarrow8$

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Example Question #121 : 3 Dimensional Space

A point in space is located, in cylindrical coordinates, at $(16,145^{\circ},23)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(7.48,14.1,23)

(9.18,-13.1,23)

(-13.1,9.18,23)

(-9.18,-13.1,23)

(14.1,7.48,23)

Correct answer:

(-13.1,9.18,23)

Explanation:

When given cylindrical coordinates of the form $(r,\theta,z)$ and converting to Cartesian coordinates of the form (x,y,z) , the relationship between , , , and $\theta$ will be of use:

$x=rcos(\theta)$

$y=rsin(\theta)$

z=z (This value is identical across Cartesian and cylindrical coordinate systems)

For the cylindrical coordinates $(16,145^{\circ},23)$

When finding the Cartesian coordinates, be sure if using a calculator to have the correct angle units set (degrees vs radians):

$x=rcos(\theta)\rightarrow-13.1$

$y=rsin(\theta)\rightarrow9.18$

$z=z\rightarrow23$

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Example Question #38 : Cylindrical Coordinates

A point in space is located, in cylindrical coordinates, at $(10,-90^{\circ},13)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(-8.94,-4.48,13)

(4.48,-8.94,13)

(0,-10,13)

(-4.48,-8.94,13)

(-10,0,13)

Correct answer:

(0,-10,13)

Explanation:

When given cylindrical coordinates of the form $(r,\theta,z)$ and converting to Cartesian coordinates of the form (x,y,z) , the relationship between , , , and $\theta$ will be of use:

$x=rcos(\theta)$

$y=rsin(\theta)$

z=z (This value is identical across Cartesian and cylindrical coordinate systems)

For the cylindrical coordinates $(10,-90^{\circ},13)$

When finding the Cartesian coordinates, be sure if using a calculator to have the correct angle units set (degrees vs radians):

$x=rcos(\theta)\rightarrow0$

$y=rsin(\theta)\rightarrow-10$

$z=z\rightarrow13$

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Example Question #39 : Cylindrical Coordinates

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

Possible Answers:

$(8.49,135^{\circ},-6)$

$(8.49,-135^{\circ},-6)$

$(12,135^{\circ},-6)$

$(8.49,-45^{\circ},-6)$

$(8.49,45^{\circ},-6)$

Correct answer:

$(8.49,45^{\circ},-6)$

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 (6,6,-6)

$\begin{align*}&r=6^2+6^2=8.49\\&\theta=arctan(\frac{6}{6})=45^{\circ}\\&z=-6\end{align*}$

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Example Question #40 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-9,7,-4) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(11.4,142.13^{\circ},-4)$

$(11.4,37.87^{\circ},-4)$

$(11.4,-142.13^{\circ},-4)$

$(-2,142.13^{\circ},-4)$

$(-2,37.87^{\circ},-4)$

Correct answer:

$(11.4,142.13^{\circ},-4)$

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 (-9,7,-4)

$\begin{align*}&r=\sqrt{(-9)^2+(7)^2}=11.4\\&\theta=arctan(\frac{7}{-9})=142.13^{\circ}\\&z=-4\end{align*}$

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Example Question #41 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (8,1,7) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(8.06,-172.87^{\circ},7)$

$(8.06,172.87^{\circ},7)$

$(8.06,-7.13^{\circ},7)$

$(9,172.87^{\circ},7)$

$(8.06,7.13^{\circ},7)$

Correct answer:

$(8.06,7.13^{\circ},7)$

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 (8,1,7)

$\begin{align*}&r=\sqrt{(8)^2+(1)^2}=8.06\\&\theta=arctan(\frac{1}{8})=7.13^{\circ}\\&z=7\end{align*}$

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Example Question #42 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (3,4,7) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(7,-53.13^{\circ},7)$

$(5,53.13^{\circ},7)$

$(7,53.13^{\circ},7)$

$(5,-126.87^{\circ},7)$

$(7,126.87^{\circ},7)$

Correct answer:

$(5,53.13^{\circ},7)$

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 (3,4,7)

$\begin{align*}&r=\sqrt{(3)^2+(4)^2}=5\\&\theta=arctan(\frac{4}{3})=53.13^{\circ}\\&z=7\end{align*}$

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Example Question #43 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-5,-7,5) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(8.6,54.46^{\circ},5)$

$(8.6,125.54^{\circ},5)$

$(-12,-125.54^{\circ},5)$

$(8.6,-125.54^{\circ},5)$

$(-12,54.46^{\circ},5)$

Correct answer:

$(8.6,-125.54^{\circ},5)$

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 (-5,-7,5)

$\begin{align*}&r=\sqrt{(-5)^2+(-7)^2}=8.6\\&\theta=arctan(\frac{-7}{-5})=-125.54^{\circ}\\&z=5\end{align*}$

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Example Question #122 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (-8,4,1) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(-4,26.57^{\circ},1)$

$(8.94,-26.57^{\circ},1)$

$(8.94,-153.43^{\circ},1)$

$(8.94,153.43^{\circ},1)$

$(-4,153.43^{\circ},1)$

Correct answer:

$(8.94,153.43^{\circ},1)$

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 (-8,4,1)

$\begin{align*}&r=\sqrt{(-8)^2+(4)^2}=8.94\\&\theta=arctan(\frac{4}{-8})=153.43^{\circ}\\&z=1\end{align*}$

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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 #35 : Cylindrical Coordinates

Example Question #36 : Cylindrical Coordinates

Example Question #121 : 3 Dimensional Space

Example Question #38 : Cylindrical Coordinates

Example Question #39 : Cylindrical Coordinates

Example Question #40 : Cylindrical Coordinates

Example Question #41 : Cylindrical Coordinates

Example Question #42 : Cylindrical Coordinates

Example Question #43 : Cylindrical Coordinates

Example Question #122 : 3 Dimensional Space

All Calculus 3 Resources

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