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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 #72 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-50,120,100) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(70,112.62^{\circ},100)$

$(130,-67.38^{\circ},100)$

$(70,67.38^{\circ},100)$

$(70,-67.38^{\circ},100)$

$(130,112.62^{\circ},100)$

Correct answer:

$(130,112.62^{\circ},100)$

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 (-50,120,100)

$\begin{align*}&r=\sqrt{(-50)^2+(120)^2}=130\\&\theta=arctan(\frac{120}{-50})=112.62^{\circ}\\&z=100\end{align*}$

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Example Question #73 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (5,-25,-6) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(25.5,78.69^{\circ},-6)$

$(20,101.31^{\circ},-6)$

$(20,-78.69^{\circ},-6)$

$(25.5,101.31^{\circ},-6)$

$(25.5,-78.69^{\circ},-6)$

Correct answer:

$(25.5,-78.69^{\circ},-6)$

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

For our coordinates (5,-25,-6)

$\begin{align*}&r=\sqrt{(5)^2+(-25)^2}=25.5\\&\theta=arctan(\frac{-25}{5})=-78.69^{\circ}\\&z=-6\end{align*}$

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Example Question #74 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (1,-1.707,6) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(0.71,120.36^{\circ},6)$

$(0.71,-120.36^{\circ},6)$

$(1.98,-59.64^{\circ},6)$

$(0.71,-59.64^{\circ},6)$

$(1.98,120.36^{\circ},6)$

Correct answer:

$(1.98,-59.64^{\circ},6)$

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

For our coordinates (1,-1.707,6)

$\begin{align*}&r=\sqrt{(1)^2+(-1.71)^2}=1.98\\&\theta=arctan(\frac{-1.71}{1})=-59.64^{\circ}\\&z=6\end{align*}$

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Example Question #75 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (7,21,22) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(22.14,108.43^{\circ},22)$

$(28,108.43^{\circ},22)$

$(28,71.57^{\circ},22)$

$(22.14,71.57^{\circ},22)$

$(22.14,-71.57^{\circ},22)$

Correct answer:

$(22.14,71.57^{\circ},22)$

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 (7,21,22)

$\begin{align*}&r=\sqrt{(7)^2+(21)^2}=22.14\\&\theta=arctan(\frac{21}{7})=71.57^{\circ}\\&z=22\end{align*}$

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Example Question #72 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (3,24,5) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(24.19,82.87^{\circ},5)$

$(27,-82.87^{\circ},5)$

$(27,97.13^{\circ},5)$

$(27,82.87^{\circ},5)$

$(24.19,97.13^{\circ},5)$

Correct answer:

$(24.19,82.87^{\circ},5)$

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 (3,24,5)

$\begin{align*}&r=\sqrt{(3)^2+(24)^2}=24.19\\&\theta=arctan(\frac{24}{3})=82.87^{\circ}\\&z=5\end{align*}$

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Example Question #73 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-2,-20,-22) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(20.1,-95.71^{\circ},-22)$

$(22,84.29^{\circ},-22)$

$(20.1,-84.29^{\circ},-22)$

$(20.1,95.71^{\circ},-22)$

$(20.1,84.29^{\circ},-22)$

Correct answer:

$(20.1,-95.71^{\circ},-22)$

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,-20,-22)

$\begin{align*}&r=\sqrt{(-2)^2+(-20)^2}=20.1\\&\theta=arctan(\frac{-20}{-2})=-95.71^{\circ}\\&z=-22\end{align*}$

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Example Question #81 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (32,-11,37) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(21,-18.97^{\circ},37)$

$(33.84,-18.97^{\circ},37)$

$(33.84,161.03^{\circ},37)$

$(21,161.03^{\circ},37)$

Correct answer:

$(33.84,-18.97^{\circ},37)$

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 (32,-11,37)

$\begin{align*}&r=\sqrt{(32)^2+(-11)^2}=33.84\\&\theta=arctan(\frac{-11}{32})=-18.97^{\circ}\\&z=37\end{align*}$

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Example Question #82 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-2,-16,-15) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(18,82.87^{\circ},-15)$

$(16.12,82.87^{\circ},-15)$

$(16.12,-97.13^{\circ},-15)$

$(18,-97.13^{\circ},-15)$

Correct answer:

$(16.12,-97.13^{\circ},-15)$

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 (-2,-16,-15)

$\begin{align*}&r=\sqrt{(-2)^2+(-16)^2}=16.12\\&\theta=arctan(\frac{-16}{-2})=-97.13^{\circ}\\&z=-15\end{align*}$

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Example Question #83 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-26,0,45) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(26,180^{\circ},45)$

$(26,45^{\circ},45)$

$(26,-45^{\circ},45)$

$(26,0^{\circ},45)$

Correct answer:

$(26,180^{\circ},45)$

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 (-26,0,45)

$\begin{align*}&r=\sqrt{(-26)^2+(0)^2}=26\\&\theta=arctan(\frac{0}{-26})=180^{\circ}\\&z=45\end{align*}$

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Example Question #84 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-8,-3,-23) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(8.54,-159.44^{\circ},-23)$

$(11,20.56^{\circ},-23)$

$(11,-159.44^{\circ},-23)$

$(8.54,20.56^{\circ},-23)$

Correct answer:

$(8.54,-159.44^{\circ},-23)$

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,-3,-23)

$\begin{align*}&r=\sqrt{(-8)^2+(-3)^2}=8.54\\&\theta=arctan(\frac{-3}{-8})=-159.44^{\circ}\\&z=-23\end{align*}$

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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 #72 : Cylindrical Coordinates

Example Question #73 : Cylindrical Coordinates

Example Question #74 : Cylindrical Coordinates

Example Question #75 : Cylindrical Coordinates

Example Question #72 : Cylindrical Coordinates

Example Question #73 : Cylindrical Coordinates

Example Question #81 : Cylindrical Coordinates

Example Question #82 : Cylindrical Coordinates

Example Question #83 : Cylindrical Coordinates

Example Question #84 : Cylindrical Coordinates

All Calculus 3 Resources

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