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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 #111 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (-9,3,9) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(90,-18.44^{\circ},9)$

$(9.49,161.57^{\circ},9)$

$(9.49,-18.44^{\circ},9)$

$(90,18.44^{\circ},9)$

$(90,161.57^{\circ},9)$

Correct answer:

$(9.49,161.57^{\circ},9)$

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 (-9,3,9)

$r=\sqrt{x^2+y^2}\rightarrow9.49$

$\theta=arctan(\frac{y}{x})\rightarrow 161.57^{\circ}$ (Bearing in mind sign convention)

z=9

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Example Question #112 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (1,-5,-6) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(-4,101.31^{\circ},-6)$

$(5.1,101.31^{\circ},-6)$

$(5.1,-78.69^{\circ},-6)$

$(-4,-101.31^{\circ},-6)$

$(5.1,78.69^{\circ},-6)$

Correct answer:

$(5.1,-78.69^{\circ},-6)$

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 (1,-5,-6)

$r=\sqrt{x^2+y^2}\rightarrow5.1$

$\theta=arctan(\frac{y}{x})\rightarrow -78.69^{\circ}$ (Bearing in mind sign convention)

z=-6

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Example Question #113 : 3 Dimensional Space

A point in space is located, in cylindrical coordinates, at $(5,90^{\circ},7)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(5,5,7)

(5,0,7)

(0,5,7)

(0,-5,7)

(-5,5,7)

Correct answer:

(0,5,7)

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

The Cartesian coordinates are $(5,90^{\circ},7)$

$x=rcos(\theta)\rightarrow0$

$y=rsin(\theta)\rightarrow5$

$z=z\rightarrow7$

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Example Question #114 : 3 Dimensional Space

A point in space is located, in cylindrical coordinates, at $(6,45^{\circ},2)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(-4.24,4.24,2)

(4.24,-4.24,2)

(-2,-2,2)

(4.24,4.24,2)

(2,2,2)

Correct answer:

(4.24,4.24,2)

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 $(6,45^{\circ},2)$

The Cartesian coordinates are

$x=rcos(\theta)\rightarrow4.24$

$y=rsin(\theta)\rightarrow4.24$

$z=z\rightarrow2$

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Example Question #29 : Cylindrical Coordinates

A point in space is located, in cylindrical coordinates, at $(9,30^{\circ},2)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(7.79,4.5,2)

(-7.79,4.5,2)

(-4.5,7.79,2)

(4.5,-7.79,2)

(4.5,7.79,2)

Correct answer:

(7.79,4.5,2)

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 $(9,30^{\circ},2)$

The Cartesian coordinates are

$x=rcos(\theta)\rightarrow7.79$

$y=rsin(\theta)\rightarrow4.5$

$z=z\rightarrow2$

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Example Question #22 : Cylindrical Coordinates

A point in space is located, in cylindrical coordinates, at $(5,60^{\circ},5)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(-1.52,-4.76,5)

(4.33,2.5,5)

(-4.76,1.52,5)

(2.5,4.33,5)

(-4.76,-1.52,5)

Correct answer:

(2.5,4.33,5)

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 $(5,60^{\circ},5)$

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

$x=rcos(\theta)\rightarrow2.5$

$y=rsin(\theta)\rightarrow4.33$

$z=z\rightarrow5$

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Example Question #1781 : Calculus 3

A point in space is located, in cylindrical coordinates, at $(10,-135^{\circ},4)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(-9.96,-0.884,4)

(-0.884,-9.96,4)

(-7.07,-7.07,4)

(7.07,7.07,4)

(-7.07,7.07,4)

Correct answer:

(-7.07,-7.07,4)

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 $(10,-135^{\circ},4)$

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

$x=rcos(\theta)\rightarrow-7.07$

$y=rsin(\theta)\rightarrow-7.07$

$z=z\rightarrow4$

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Example Question #32 : Cylindrical Coordinates

A point in space is located, in cylindrical coordinates, at $(12,120^{\circ},14)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(10.4,-6,14)

(-6,10.4,14)

(6.97,9.77,14)

(-9.77,6.97,14)

(9.77,6.97,14)

Correct answer:

(-6,10.4,14)

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 $(12,120^{\circ},14)$

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

$x=rcos(\theta)\rightarrow-6$

$y=rsin(\theta)\rightarrow10.4$

$z=z\rightarrow14$

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Example Question #33 : Cylindrical Coordinates

A point in space is located, in cylindrical coordinates, at $(15,20^{\circ},11)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(14.1,5.13,11)

(5.13,14.1,11)

(-6.12,-13.7,11)

(6.12,13.7,11)

(13.7,6.12,11)

Correct answer:

(14.1,5.13,11)

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 $(15,20^{\circ},11)$

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

$x=rcos(\theta)\rightarrow14.1$

$y=rsin(\theta)\rightarrow5.13$

$z=z\rightarrow11$

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Example Question #34 : Cylindrical Coordinates

A point in space is located, in cylindrical coordinates, at $(20,180^{\circ},5)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(-20,0,5)

(0,-20,5)

(20,0,5)

(0,20,5)

(-12,-16,5)

Correct answer:

(-20,0,5)

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 $(20,180^{\circ},5)$

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

$x=rcos(\theta)\rightarrow-20$

$y=rsin(\theta)\rightarrow0$

$z=z\rightarrow5$

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 #111 : 3 Dimensional Space

Example Question #112 : 3 Dimensional Space

Example Question #113 : 3 Dimensional Space

Example Question #114 : 3 Dimensional Space

Example Question #29 : Cylindrical Coordinates

Example Question #22 : Cylindrical Coordinates

Example Question #1781 : Calculus 3

Example Question #32 : Cylindrical Coordinates

Example Question #33 : Cylindrical Coordinates

Example Question #34 : Cylindrical Coordinates

All Calculus 3 Resources

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