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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 #41 : Cylindrical Coordinates

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

Possible Answers:

$(-5,-165.96^{\circ},-4)$

$(4.12,165.96^{\circ},-4)$

$(4.12,-165.96^{\circ},-4)$

$(4.12,14.04^{\circ},-4)$

$(-5,14.04^{\circ},-4)$

Correct answer:

$(4.12,-165.96^{\circ},-4)$

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

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 (-4,-1,-4)

$\begin{align*}&r=\sqrt{(-4)^2+(-1)^2}=4.12\\&\theta=arctan(\frac{-1}{-4})=-165.96^{\circ}\\&z=-4\end{align*}$

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Example Question #42 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (3,6,-5) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(9,63.43^{\circ},-5)$

$(9,116.57^{\circ},-5)$

$(6.71,-116.57^{\circ},-5)$

$(6.71,116.57^{\circ},-5)$

$(6.71,63.43^{\circ},-5)$

Correct answer:

$(6.71,63.43^{\circ},-5)$

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 (3,6,-5)

$\begin{align*}&r=\sqrt{(3)^2+(6)^2}=6.71\\&\theta=arctan(\frac{6}{3})=63.43^{\circ}\\&z=-5\end{align*}$

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Example Question #42 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-6,-6,-4) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(8.49,-135^{\circ},-4)$

$(12,135^{\circ},-4)$

$(8.49,-45^{\circ},-4)$

$(12,45^{\circ},-4)$

$(8.49,45^{\circ},-4)$

Correct answer:

$(8.49,-135^{\circ},-4)$

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 (-6,-6,-4)

$\begin{align*}&r=\sqrt{(-6)^2+(-6)^2}=8.49\\&\theta=arctan(\frac{-6}{-6})=-135^{\circ}\\&z=-4\end{align*}$

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Example Question #43 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (9,-1,4) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(8,-6.34^{\circ},4)$

$(8,173.66^{\circ},4)$

$(9.06,173.66^{\circ},4)$

$(8,-173.66^{\circ},4)$

$(9.06,-6.34^{\circ},4)$

Correct answer:

$(9.06,-6.34^{\circ},4)$

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,-1,4)

$\begin{align*}&r=\sqrt{(9)^2+(-1)^2}=9.06\\&\theta=arctan(\frac{-1}{9})=-6.34^{\circ}\\&z=4\end{align*}$

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Example Question #44 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (4,2,-5) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(4.47,-153.43^{\circ},-5)$

$(4.47,-26.57^{\circ},-5)$

$(4.47,26.57^{\circ},-5)$

$(6,153.43^{\circ},-5)$

$(6,26.57^{\circ},-5)$

Correct answer:

$(4.47,26.57^{\circ},-5)$

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 (4,2,-5)

$\begin{align*}&r=\sqrt{(4)^2+(2)^2}=4.47\\&\theta=arctan(\frac{2}{4})=26.57^{\circ}\\&z=-5\end{align*}$

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Example Question #46 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-7,-9,5) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(11.4,-52.13^{\circ},5)$

$(16,-127.87^{\circ},5)$

$(16,52.13^{\circ},5)$

$(11.4,52.13^{\circ},5)$

$(11.4,-127.87^{\circ},5)$

Correct answer:

$(11.4,-127.87^{\circ},5)$

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 (-7,-9,5)

$\begin{align*}&r=\sqrt{(-7)^2+(-9)^2}=11.4\\&\theta=arctan(\frac{-9}{-7})=-127.87^{\circ}\\&z=5\end{align*}$

Report an Error

Example Question #51 : Cylindrical Coordinates

A point in space is located, in cylindrical coordinates, at $(3,25^{\circ},-8)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(1.27,2.72,-8)

(2.97,0.4,-8)

(-0.4,2.97,-8)

(2.72,1.27,-8)

(2.97,-0.4,-8)

Correct answer:

(2.72,1.27,-8)

Explanation:

If asked to convert cylindrical coordinates of the form $(r,\theta,z)$ to Cartesian coordinates of the form (x,y,z) , it is necessary to relate and to the radius, , and the angle, $\theta$ . The relationships are as follows:

$x=rcos(\theta)$

$y=rsin(\theta)$

Finding is much simpler; it does not change between Cartesian and cylindrical coordinates:

z=z

However, care should be taken when finding and ; if using a calculator, it is imperative that the correct units (degrees or radians) are specified for the input!

For our coordinates $(3,25^{\circ},-8)$

$\begin{align*}&x=3cos(25^{\circ})=2.72\\&y=3sin(25^{\circ})=1.27\\&z=-8\end{align*}$

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Example Question #131 : 3 Dimensional Space

A point in space is located, in cylindrical coordinates, at $(18,175^{\circ},15)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(-14.42,10.77,15)

(17.93,1.57,15)

(1.57,-17.93,15)

(10.77,-14.42,15)

(-17.93,1.57,15)

Correct answer:

(-17.93,1.57,15)

Explanation:

$x=rcos(\theta)$

$y=rsin(\theta)$

Finding is much simpler; it does not change between Cartesian and cylindrical coordinates:

z=z

However, care should be taken when finding and ; if using a calculator, it is imperative that the correct units (degrees or radians) are specified for the input!

For our coordinates $(18,175^{\circ},15)$

$\begin{align*}&x=18cos(175^{\circ})=-17.93\\&y=18sin(175^{\circ})=1.57\\&z=15\end{align*}$

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Example Question #53 : Cylindrical Coordinates

A point in space is located, in cylindrical coordinates, at $(10,60^{\circ},-1)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(8.66,5,-1)

(-3.05,-9.52,-1)

(9.52,3.05,-1)

(5,8.66,-1)

(-9.52,-3.05,-1)

Correct answer:

(5,8.66,-1)

Explanation:

$x=rcos(\theta)$

$y=rsin(\theta)$

Finding is much simpler; it does not change between Cartesian and cylindrical coordinates:

z=z

However, care should be taken when finding and ; if using a calculator, it is imperative that the correct units (degrees or radians) are specified for the input!

For our coordinates $(10,60^{\circ},-1)$

$\begin{align*}&x=10cos(60^{\circ})=5\\&y=10sin(60^{\circ})=8.66\\&z=-1\end{align*}$

Report an Error

Example Question #132 : 3 Dimensional Space

A point in space is located, in cylindrical coordinates, at $(10,115^{\circ},12)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(9.45,3.26,12)

(-4.23,9.06,12)

(9.06,-4.23,12)

(9.45,-3.26,12)

(-3.26,9.45,12)

Correct answer:

(-4.23,9.06,12)

Explanation:

$x=rcos(\theta)$

$y=rsin(\theta)$

Finding is much simpler; it does not change between Cartesian and cylindrical coordinates:

z=z

However, care should be taken when finding and ; if using a calculator, it is imperative that the correct units (degrees or radians) are specified for the input!

For our coordinates $(10,115^{\circ},12)$

$\begin{align*}&x=10cos(115^{\circ})=-4.23\\&y=10sin(115^{\circ})=9.06\\&z=12\end{align*}$

Report an Error

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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 #41 : Cylindrical Coordinates

Example Question #42 : Cylindrical Coordinates

Example Question #42 : Cylindrical Coordinates

Example Question #43 : Cylindrical Coordinates

Example Question #44 : Cylindrical Coordinates

Example Question #46 : Cylindrical Coordinates

Example Question #51 : Cylindrical Coordinates

Example Question #131 : 3 Dimensional Space

Example Question #53 : Cylindrical Coordinates

Example Question #132 : 3 Dimensional Space

All Calculus 3 Resources

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