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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 #61 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-8,-5,-2) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(13,-147.99^{\circ},-2)$

$(13,32.01^{\circ},-2)$

$(9.43,-147.99^{\circ},-2)$

$(9.43,147.99^{\circ},-2)$

$(9.43,32.01^{\circ},-2)$

Correct answer:

$(9.43,-147.99^{\circ},-2)$

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 (-8,-5,-2)

$\begin{align*}&r=\sqrt{(-8)^2+(-5)^2}=9.43\\&\theta=arctan(\frac{-5}{-8})=-147.99^{\circ}\\&z=-2\end{align*}$

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Example Question #66 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-14,-18,-10) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(32,127.87^{\circ},-10)$

$(22.8,-52.13^{\circ},-10)$

$(22.8,-127.87^{\circ},-10)$

$(32,52.13^{\circ},-10)$

$(22.8,52.13^{\circ},-10)$

Correct answer:

$(22.8,-127.87^{\circ},-10)$

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 (-14,-18,-10)

$\begin{align*}&r=\sqrt{(-14)^2+(-18)^2}=22.8\\&\theta=arctan(\frac{-18}{-14})=-127.87^{\circ}\\&z=-10\end{align*}$

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Example Question #67 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (10,14,-10) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(17.2,-125.54^{\circ},-10)$

$(24,125.54^{\circ},-10)$

$(17.2,125.54^{\circ},-10)$

$(17.2,-54.46^{\circ},-10)$

$(17.2,54.46^{\circ},-10)$

Correct answer:

$(17.2,54.46^{\circ},-10)$

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 (10,14,-10)

$\begin{align*}&r=\sqrt{(10)^2+(14)^2}=17.2\\&\theta=arctan(\frac{14}{10})=54.46^{\circ}\\&z=-10\end{align*}$

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Example Question #68 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (0,16,20) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(16,-90^{\circ},20)$

$(16,135^{\circ},20)$

$(16,90^{\circ},20)$

$(16,-135^{\circ},20)$

$(16,45^{\circ},20)$

Correct answer:

$(16,90^{\circ},20)$

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 (0,16,20)

$\begin{align*}&r=\sqrt{(0)^2+(16)^2}=16\\&\theta=arctan(\frac{16}{0})=90^{\circ}\\&z=20\end{align*}$

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Example Question #69 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (12,-5,31) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(13,-22.62^{\circ},31)$

$(13,157.38^{\circ},31)$

$(13,22.62^{\circ},31)$

$(7,157.38^{\circ},31)$

$(7,-22.62^{\circ},31)$

Correct answer:

$(13,-22.62^{\circ},31)$

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,-5,31)

$\begin{align*}&r=\sqrt{(12)^2+(-5)^2}=13\\&\theta=arctan(\frac{-5}{12})=-22.62^{\circ}\\&z=31\end{align*}$

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Example Question #70 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-24,10,32) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(26,22.62^{\circ},32)$

$(26,-22.62^{\circ},32)$

$(26,157.38^{\circ},32)$

$(14,22.62^{\circ},32)$

$(14,157.38^{\circ},32)$

Correct answer:

$(26,157.38^{\circ},32)$

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 (-24,10,32)

$\begin{align*}&r=\sqrt{(-24)^2+(10)^2}=26\\&\theta=arctan(\frac{10}{-24})=157.38^{\circ}\\&z=32\end{align*}$

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Example Question #71 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (20,14,25) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(34,145.01^{\circ},25)$

$(34,34.99^{\circ},25)$

$(34,-145.01^{\circ},25)$

$(24.41,145.01^{\circ},25)$

$(24.41,34.99^{\circ},25)$

Correct answer:

$(24.41,34.99^{\circ},25)$

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 (20,14,25)

$\begin{align*}&r=\sqrt{(20)^2+(14)^2}=24.41\\&\theta=arctan(\frac{14}{20})=34.99^{\circ}\\&z=25\end{align*}$

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Example Question #71 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-9,12,15) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(15,-53.13^{\circ},15)$

$(15,53.13^{\circ},15)$

$(3,126.87^{\circ},15)$

$(15,126.87^{\circ},15)$

$(3,-53.13^{\circ},15)$

Correct answer:

$(15,126.87^{\circ},15)$

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 (-9,12,15)

$\begin{align*}&r=\sqrt{(-9)^2+(12)^2}=15\\&\theta=arctan(\frac{12}{-9})=126.87^{\circ}\\&z=15\end{align*}$

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Example Question #73 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (36,-160,18) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(164,102.68^{\circ},18)$

$(164,-77.32^{\circ},18)$

$(124,102.68^{\circ},18)$

$(124,-77.32^{\circ},18)$

$(124,77.32^{\circ},18)$

Correct answer:

$(164,-77.32^{\circ},18)$

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 (36,-160,18)

$\begin{align*}&r=\sqrt{(36)^2+(-160)^2}=164\\&\theta=arctan(\frac{-160}{36})=-77.32^{\circ}\\&z=18\end{align*}$

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Example Question #74 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-1.5,-2,10) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(2.5,53.13^{\circ},10)$

$(2.5,-126.87^{\circ},10)$

$(3.5,-126.87^{\circ},10)$

$(2.5,126.87^{\circ},10)$

$(3.5,53.13^{\circ},10)$

Correct answer:

$(2.5,-126.87^{\circ},10)$

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 (-1.5,-2,10)

$\begin{align*}&r=\sqrt{(-1.5)^2+(-2)^2}=2.5\\&\theta=arctan(\frac{-2}{-1.5})=-126.87^{\circ}\\&z=10\end{align*}$

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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 #61 : Cylindrical Coordinates

Example Question #66 : Cylindrical Coordinates

Example Question #67 : Cylindrical Coordinates

Example Question #68 : Cylindrical Coordinates

Example Question #69 : Cylindrical Coordinates

Example Question #70 : Cylindrical Coordinates

Example Question #71 : Cylindrical Coordinates

Example Question #71 : Cylindrical Coordinates

Example Question #73 : Cylindrical Coordinates

Example Question #74 : Cylindrical Coordinates

All Calculus 3 Resources

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