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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 #158 : Spherical Coordinates

$\begin{align*}&\text{A point in space is located, in spherical coordinates, at } (19,-94^{\circ},33^{\circ})\text{ of form: } (\rho,\theta,\phi)\\&\text{Where }\phi\text{ is the angle between the vector, }\rho\text{, to the point and the }z\text{ axis}.\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(18.42,4.66,-0.25)

(-1.11,-15.9,10.35)

(-0.72,-10.32,15.93)

(-0.24,-0.06,19)

Correct answer:

(-0.72,-10.32,15.93)

Explanation:

$\begin{align*}&\text{When converting spherical coordinates of the form} (\rho,\theta,\phi)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{, }y\text{, and }z\\& \text{to the vector, }\rho\text{, and the angles }\theta.\text{ and }\phi.\text{ The relationships are as follows:}\\&x=\rho sin(\phi)cos(\theta)\\&y=\rho sin(\phi)sin(\theta)\\&z=\rho cos(\phi)\\&\text{You may notice that }\rho sin(\phi)\text{ is the projection of }\rho\text{ on the }xy\text{-plane!}\\&\text{However, care should be taken when finding these terms}\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=19sin(33^{\circ})cos(-94^{\circ})=-0.72\\&y=19sin(33^{\circ})sin(-94^{\circ})=-10.32\\&z=19cos(33^{\circ})15.93\end{align*}$

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Example Question #159 : Spherical Coordinates

$\begin{align*}&\text{A point in space is located, in spherical coordinates, at } (142,-4^{\circ},88^{\circ})\text{ of form: } (\rho,\theta,\phi)\\&\text{Where }\phi\text{ is the angle between the vector, }\rho\text{, to the point and the }z\text{ axis}.\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-3.29,3.8,141.91)

(-92.76,107.4,5.03)

(141.57,-9.9,4.96)

(4.94,-0.35,141.91)

Correct answer:

(141.57,-9.9,4.96)

Explanation:

$\begin{align*}&\text{When converting spherical coordinates of the form} (\rho,\theta,\phi)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{, }y\text{, and }z\\& \text{to the vector, }\rho\text{, and the angles }\theta.\text{ and }\phi.\text{ The relationships are as follows:}\\&x=\rho sin(\phi)cos(\theta)\\&y=\rho sin(\phi)sin(\theta)\\&z=\rho cos(\phi)\\&\text{You may notice that }\rho sin(\phi)\text{ is the projection of }\rho\text{ on the }xy\text{-plane!}\\&\text{However, care should be taken when finding these terms}\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=142sin(88^{\circ})cos(-4^{\circ})=141.57\\&y=142sin(88^{\circ})sin(-4^{\circ})=-9.9\\&z=142cos(88^{\circ})4.96\end{align*}$

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Example Question #160 : Spherical Coordinates

$\begin{align*}&\text{A point in space is located, in spherical coordinates, at } (51,-47^{\circ},162^{\circ})\text{ of form: } (\rho,\theta,\phi)\\&\text{Where }\phi\text{ is the angle between the vector, }\rho\text{, to the point and the }z\text{ axis}.\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-33.08,35.47,15.76)

(-10.45,-1.3,-49.9)

(10.75,-11.53,-48.5)

(49.52,6.17,10.53)

Correct answer:

(10.75,-11.53,-48.5)

Explanation:

$\begin{align*}&\text{When converting spherical coordinates of the form} (\rho,\theta,\phi)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{, }y\text{, and }z\\& \text{to the vector, }\rho\text{, and the angles }\theta.\text{ and }\phi.\text{ The relationships are as follows:}\\&x=\rho sin(\phi)cos(\theta)\\&y=\rho sin(\phi)sin(\theta)\\&z=\rho cos(\phi)\\&\text{You may notice that }\rho sin(\phi)\text{ is the projection of }\rho\text{ on the }xy\text{-plane!}\\&\text{However, care should be taken when finding these terms}\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=51sin(162^{\circ})cos(-47^{\circ})=10.75\\&y=51sin(162^{\circ})sin(-47^{\circ})=-11.53\\&z=51cos(162^{\circ})-48.5\end{align*}$

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Example Question #161 : Spherical Coordinates

$\begin{align*}&\text{A point in space is located, in spherical coordinates, at } (17,-40^{\circ},140^{\circ})\text{ of form: } (\rho,\theta,\phi)\\&\text{Where }\phi\text{ is the angle between the vector, }\rho\text{, to the point and the }z\text{ axis}.\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-11.11,-12.42,-3.36)

(-9.98,8.37,10.93)

(2.24,2.51,16.66)

(8.37,-7.02,-13.02)

Correct answer:

(8.37,-7.02,-13.02)

Explanation:

$\begin{align*}&\text{When converting spherical coordinates of the form} (\rho,\theta,\phi)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{, }y\text{, and }z\\& \text{to the vector, }\rho\text{, and the angles }\theta.\text{ and }\phi.\text{ The relationships are as follows:}\\&x=\rho sin(\phi)cos(\theta)\\&y=\rho sin(\phi)sin(\theta)\\&z=\rho cos(\phi)\\&\text{You may notice that }\rho sin(\phi)\text{ is the projection of }\rho\text{ on the }xy\text{-plane!}\\&\text{However, care should be taken when finding these terms}\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=17sin(140^{\circ})cos(-40^{\circ})=8.37\\&y=17sin(140^{\circ})sin(-40^{\circ})=-7.02\\&z=17cos(140^{\circ})-13.02\end{align*}$

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Example Question #162 : Spherical Coordinates

$\begin{align*}&\text{A point in space is located, in spherical coordinates, at } (37,-146^{\circ},73^{\circ})\text{ of form: } (\rho,\theta,\phi)\\&\text{Where }\phi\text{ is the angle between the vector, }\rho\text{, to the point and the }z\text{ axis}.\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-2.1,24.95,-27.24)

(-8.97,-6.05,35.38)

(-29.33,-19.79,10.82)

(-2.29,27.14,-25.04)

Correct answer:

(-29.33,-19.79,10.82)

Explanation:

$\begin{align*}&\text{When converting spherical coordinates of the form} (\rho,\theta,\phi)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{, }y\text{, and }z\\& \text{to the vector, }\rho\text{, and the angles }\theta.\text{ and }\phi.\text{ The relationships are as follows:}\\&x=\rho sin(\phi)cos(\theta)\\&y=\rho sin(\phi)sin(\theta)\\&z=\rho cos(\phi)\\&\text{You may notice that }\rho sin(\phi)\text{ is the projection of }\rho\text{ on the }xy\text{-plane!}\\&\text{However, care should be taken when finding these terms}\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=37sin(73^{\circ})cos(-146^{\circ})=-29.33\\&y=37sin(73^{\circ})sin(-146^{\circ})=-19.79\\&z=37cos(73^{\circ})10.82\end{align*}$

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Example Question #501 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in spherical coordinates, at } (20,165^{\circ},169^{\circ})\text{ of form: } (\rho,\theta,\phi)\\&\text{Where }\phi\text{ is the angle between the vector, }\rho\text{, to the point and the }z\text{ axis}.\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(0.8,-12.01,15.97)

(-1.06,15.93,-12.04)

(18.96,-5.08,3.82)

(-3.69,0.99,-19.63)

Correct answer:

(-3.69,0.99,-19.63)

Explanation:

$\begin{align*}&\text{When converting spherical coordinates of the form} (\rho,\theta,\phi)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{, }y\text{, and }z\\& \text{to the vector, }\rho\text{, and the angles }\theta.\text{ and }\phi.\text{ The relationships are as follows:}\\&x=\rho sin(\phi)cos(\theta)\\&y=\rho sin(\phi)sin(\theta)\\&z=\rho cos(\phi)\\&\text{You may notice that }\rho sin(\phi)\text{ is the projection of }\rho\text{ on the }xy\text{-plane!}\\&\text{However, care should be taken when finding these terms}\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=20sin(169^{\circ})cos(165^{\circ})=-3.69\\&y=20sin(169^{\circ})sin(165^{\circ})=0.99\\&z=20cos(169^{\circ})-19.63\end{align*}$

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Example Question #164 : Spherical Coordinates

$\begin{align*}&\text{A point in space is located, in spherical coordinates, at } (87,-96^{\circ},11^{\circ})\text{ of form: } (\rho,\theta,\phi)\\&\text{Where }\phi\text{ is the angle between the vector, }\rho\text{, to the point and the }z\text{ axis}.\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-1.74,-16.51,85.4)

(-0.07,-0.38,-87)

(15.7,85.57,0.39)

(-8.93,-84.93,16.6)

Correct answer:

(-1.74,-16.51,85.4)

Explanation:

$\begin{align*}&\text{When converting spherical coordinates of the form} (\rho,\theta,\phi)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{, }y\text{, and }z\\& \text{to the vector, }\rho\text{, and the angles }\theta.\text{ and }\phi.\text{ The relationships are as follows:}\\&x=\rho sin(\phi)cos(\theta)\\&y=\rho sin(\phi)sin(\theta)\\&z=\rho cos(\phi)\\&\text{You may notice that }\rho sin(\phi)\text{ is the projection of }\rho\text{ on the }xy\text{-plane!}\\&\text{However, care should be taken when finding these terms}\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=87sin(11^{\circ})cos(-96^{\circ})=-1.74\\&y=87sin(11^{\circ})sin(-96^{\circ})=-16.51\\&z=87cos(11^{\circ})85.4\end{align*}$

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Example Question #165 : Spherical Coordinates

$\begin{align*}&\text{A point in space is located, in spherical coordinates, at } (53,-175^{\circ},147^{\circ})\text{ of form: } (\rho,\theta,\phi)\\&\text{Where }\phi\text{ is the angle between the vector, }\rho\text{, to the point and the }z\text{ axis}.\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-25.16,-33.68,32.28)

(-28.76,-2.52,-44.45)

(44.28,3.87,28.87)

(19.32,25.86,-42.04)

Correct answer:

(-28.76,-2.52,-44.45)

Explanation:

$\begin{align*}&\text{When converting spherical coordinates of the form} (\rho,\theta,\phi)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{, }y\text{, and }z\\& \text{to the vector, }\rho\text{, and the angles }\theta.\text{ and }\phi.\text{ The relationships are as follows:}\\&x=\rho sin(\phi)cos(\theta)\\&y=\rho sin(\phi)sin(\theta)\\&z=\rho cos(\phi)\\&\text{You may notice that }\rho sin(\phi)\text{ is the projection of }\rho\text{ on the }xy\text{-plane!}\\&\text{However, care should be taken when finding these terms}\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=53sin(147^{\circ})cos(-175^{\circ})=-28.76\\&y=53sin(147^{\circ})sin(-175^{\circ})=-2.52\\&z=53cos(147^{\circ})-44.45\end{align*}$

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Example Question #161 : Spherical Coordinates

$\begin{align*}&\text{A point in space is located, in spherical coordinates, at } (7,54^{\circ},31^{\circ})\text{ of form: } (\rho,\theta,\phi)\\&\text{Where }\phi\text{ is the angle between the vector, }\rho\text{, to the point and the }z\text{ axis}.\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-5.31,-3.58,-2.83)

(2.12,2.92,6)

(2.35,1.58,6.4)

(3.53,4.85,3.61)

Correct answer:

(2.12,2.92,6)

Explanation:

$\begin{align*}&\text{When converting spherical coordinates of the form} (\rho,\theta,\phi)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{, }y\text{, and }z\\& \text{to the vector, }\rho\text{, and the angles }\theta.\text{ and }\phi.\text{ The relationships are as follows:}\\&x=\rho sin(\phi)cos(\theta)\\&y=\rho sin(\phi)sin(\theta)\\&z=\rho cos(\phi)\\&\text{You may notice that }\rho sin(\phi)\text{ is the projection of }\rho\text{ on the }xy\text{-plane!}\\&\text{However, care should be taken when finding these terms}\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=7sin(31^{\circ})cos(54^{\circ})=2.12\\&y=7sin(31^{\circ})sin(54^{\circ})=2.92\\&z=7cos(31^{\circ})6\end{align*}$

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Example Question #167 : Spherical Coordinates

$\begin{align*}&\text{A point in space is located, in spherical coordinates, at } (110,-18^{\circ},116^{\circ})\text{ of form: } (\rho,\theta,\phi)\\&\text{Where }\phi\text{ is the angle between the vector, }\rho\text{, to the point and the }z\text{ axis}.\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(94.03,-30.55,-48.22)

(-70.57,-80.26,26.03)

(17.19,19.55,-106.88)

(-45.86,14.9,98.87)

Correct answer:

(94.03,-30.55,-48.22)

Explanation:

$\begin{align*}&\text{When converting spherical coordinates of the form} (\rho,\theta,\phi)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{, }y\text{, and }z\\& \text{to the vector, }\rho\text{, and the angles }\theta.\text{ and }\phi.\text{ The relationships are as follows:}\\&x=\rho sin(\phi)cos(\theta)\\&y=\rho sin(\phi)sin(\theta)\\&z=\rho cos(\phi)\\&\text{You may notice that }\rho sin(\phi)\text{ is the projection of }\rho\text{ on the }xy\text{-plane!}\\&\text{However, care should be taken when finding these terms}\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=110sin(116^{\circ})cos(-18^{\circ})=94.03\\&y=110sin(116^{\circ})sin(-18^{\circ})=-30.55\\&z=110cos(116^{\circ})-48.22\end{align*}$

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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 #158 : Spherical Coordinates

Example Question #159 : Spherical Coordinates

Example Question #160 : Spherical Coordinates

Example Question #161 : Spherical Coordinates

Example Question #162 : Spherical Coordinates

Example Question #501 : 3 Dimensional Space

Example Question #164 : Spherical Coordinates

Example Question #165 : Spherical Coordinates

Example Question #161 : Spherical Coordinates

Example Question #167 : Spherical Coordinates

All Calculus 3 Resources

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