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Calculus 3 : 3-Dimensional Space

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Example Questions

Example Question #141 : 3 Dimensional Space

A point in space is located, in cylindrical coordinates, at $(25,100^{\circ},14)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(-12.66,21.56,14)

(24.62,-4.34,14)

(21.56,-12.66,14)

(-4.34,24.62,14)

(12.66,21.56,14)

Correct answer:

(-4.34,24.62,14)

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 $(25,100^{\circ},14)$

$\begin{align*}&x=25cos(100^{\circ})=-4.34\\&y=25sin(100^{\circ})=24.62\\&z=14\end{align*}$

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Example Question #142 : 3 Dimensional Space

A point in space is located, in cylindrical coordinates, at $(20,10^{\circ},5)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(10.88,16.78,5)

(-16.78,-10.88,5)

(19.7,3.47,5)

(3.47,19.7,5)

Correct answer:

(19.7,3.47,5)

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 $(20,10^{\circ},5)$

$\begin{align*}&x=20cos(10^{\circ})=19.7\\&y=20sin(10^{\circ})=3.47\\&z=5\end{align*}$

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Example Question #143 : 3 Dimensional Space

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

Possible Answers:

(3.88,14.49,15)

(-11.4,9.75,15)

(9.75,-11.4,15)

(14.49,3.88,15)

(-14.49,-3.88,15)

Correct answer:

(14.49,3.88,15)

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

$\begin{align*}&x=15cos(15^{\circ})=14.49\\&y=15sin(15^{\circ})=3.88\\&z=15\end{align*}$

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Example Question #144 : 3 Dimensional Space

A point in space is located, in cylindrical coordinates, at $(10,70^{\circ},-4)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(3.42,9.4,-4)

(9.4,3.42,4)

(9.4,3.42,-4)

(6.33,7.74,-4)

(7.74,6.33,-4)

Correct answer:

(3.42,9.4,-4)

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 $(10,70^{\circ},-4)$

$\begin{align*}&x=10cos(70^{\circ})=3.42\\&y=10sin(70^{\circ})=9.4\\&z=-4\end{align*}$

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Example Question #145 : 3 Dimensional Space

A point in space is located, in cylindrical coordinates, at $(20,63^{\circ},12)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(3.35,19.72,12)

(-3.35,-19.72,12)

(9.08,17.82,12)

(17.82,9.08,12)

(19.72,3.35,12)

Correct answer:

(9.08,17.82,12)

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 $(20,63^{\circ},12)$

$\begin{align*}&x=20cos(63^{\circ})=9.08\\&y=20sin(63^{\circ})=17.82\\&z=12\end{align*}$

Report an Error

Example Question #146 : 3 Dimensional Space

A point in space is located, in cylindrical coordinates, at $(15,155^{\circ},30)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(-13.59,6.34,30)

(6.34,13.59,30)

(-13.1,-7.31,30)

(6.34,-13.59,30)

(-7.31,-13.1,30)

Correct answer:

(-13.59,6.34,30)

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 $(15,155^{\circ},30)$

$\begin{align*}&x=15cos(155^{\circ})=-13.59\\&y=15sin(155^{\circ})=6.34\\&z=30\end{align*}$

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Example Question #61 : Cylindrical Coordinates

A point in space is located, in cylindrical coordinates, at $(10,-120^{\circ},25)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(-5.81,8.14,25)

(5,8.66,25)

(-8.66,-5,25)

(8.14,-5.81,25)

(-5,-8.66,25)

Correct answer:

(-5,-8.66,25)

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 $(10,-120^{\circ},25)$

$\begin{align*}&x=10cos(-120^{\circ})=-5\\&y=10sin(-120^{\circ})=-8.66\\&z=25\end{align*}$

Report an Error

Example Question #62 : Cylindrical Coordinates

A point in space is located, in cylindrical coordinates, at $(30,-15^{\circ},4)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(-7.76,28.98,4)

(28.98,7.76,4)

(-22.79,-19.51,4)

(28.98,-7.76,4)

(-19.51,-22.79,4)

Correct answer:

(28.98,-7.76,4)

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 $(30,-15^{\circ},4)$

$\begin{align*}&x=30cos(-15^{\circ})=28.98\\&y=30sin(-15^{\circ})=-7.76\\&z=4\end{align*}$

Report an Error

Example Question #63 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (3,-13,8) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(13.34,77.01^{\circ},8)$

$(10,-77.01^{\circ},8)$

$(13.34,-77.01^{\circ},8)$

$(13.34,102.99^{\circ},8)$

$(10,102.99^{\circ},8)$

Correct answer:

$(13.34,-77.01^{\circ},8)$

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 (3,-13,8)

$\begin{align*}&r=\sqrt{(3)^2+(-13)^2}=13.34\\&\theta=arctan(\frac{-13}{3})=-77.01^{\circ}\\&z=8\end{align*}$

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Example Question #64 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (17,18,-20) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(35,46.64^{\circ},-20)$

$(24.76,46.64^{\circ},-20)$

$(24.76,133.36^{\circ},-20)$

$(24.76,-133.36^{\circ},-20)$

$(35,133.36^{\circ},-20)$

Correct answer:

$(24.76,46.64^{\circ},-20)$

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 (17,18,-20)

$\begin{align*}&r=\sqrt{(17)^2+(18)^2}=24.76\\&\theta=arctan(\frac{18}{17})=46.64^{\circ}\\&z=-20\end{align*}$

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 #141 : 3 Dimensional Space

Example Question #142 : 3 Dimensional Space

Example Question #143 : 3 Dimensional Space

Example Question #144 : 3 Dimensional Space

Example Question #145 : 3 Dimensional Space

Example Question #146 : 3 Dimensional Space

Example Question #61 : Cylindrical Coordinates

Example Question #62 : Cylindrical Coordinates

Example Question #63 : Cylindrical Coordinates

Example Question #64 : Cylindrical Coordinates

All Calculus 3 Resources

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