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Calculus 3 : 3-Dimensional Space

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Example Questions

Example Question #131 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (123,34^{\circ},-72).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(65.08,-104.37,-72)

(-104.37,65.08,-72)

(101.97,68.78,-72)

(68.78,101.97,-72)

Correct answer:

(101.97,68.78,-72)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=123cos(34^{\circ})=101.97\\&y=123sin(34^{\circ})=68.78\\&z=-72\end{align*}$

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Example Question #222 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (4,-68^{\circ},-22).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-3.71,1.5,-22)

(1.76,3.59,-22)

(3.59,1.76,-22)

(1.5,-3.71,-22)

Correct answer:

(1.5,-3.71,-22)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=4cos(-68^{\circ})=1.5\\&y=4sin(-68^{\circ})=-3.71\\&z=-22\end{align*}$

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Example Question #131 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (25,-28^{\circ},-97).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-11.74,22.07,-97)

(-24.07,-6.77,-97)

(22.07,-11.74,-97)

(-6.77,-24.07,-97)

Correct answer:

(22.07,-11.74,-97)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=25cos(-28^{\circ})=22.07\\&y=25sin(-28^{\circ})=-11.74\\&z=-97\end{align*}$

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Example Question #224 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (15,-10^{\circ},30).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(14.77,-2.6,30)

(-12.59,8.16,30)

(8.16,-12.59,30)

(-2.6,14.77,30)

Correct answer:

(14.77,-2.6,30)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=15cos(-10^{\circ})=14.77\\&y=15sin(-10^{\circ})=-2.6\\&z=30\end{align*}$

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Example Question #132 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (105,50^{\circ},60).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(101.32,-27.55,60)

(-27.55,101.32,60)

(67.49,80.43,60)

(80.43,67.49,60)

Correct answer:

(67.49,80.43,60)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=105cos(50^{\circ})=67.49\\&y=105sin(50^{\circ})=80.43\\&z=60\end{align*}$

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Example Question #133 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (141,25^{\circ},-111).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(59.59,127.79,-111)

(127.79,59.59,-111)

(-18.66,139.76,-111)

(139.76,-18.66,-111)

Correct answer:

(127.79,59.59,-111)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=141cos(25^{\circ})=127.79\\&y=141sin(25^{\circ})=59.59\\&z=-111\end{align*}$

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Example Question #141 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (95,122,-112).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(154.63,52.09^{\circ},-112)$

$(217,52.09^{\circ},-112)$

$(154.63,127.91^{\circ},-112)$

$(217,127.91^{\circ},-112)$

Correct answer:

$(154.63,52.09^{\circ},-112)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(95,122,-112)\\&r=\sqrt{(95)^2+(122)^2}=154.63\\&\theta=arctan(\frac{122}{95})=52.09^{\circ}\\&z=-112\end{align*}$

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Example Question #142 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (124,40,-121).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(164,162.12^{\circ},-121)$

$(164,17.88^{\circ},-121)$

$(130.29,17.88^{\circ},-121)$

$(130.29,162.12^{\circ},-121)$

Correct answer:

$(130.29,17.88^{\circ},-121)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(124,40,-121)\\&r=\sqrt{(124)^2+(40)^2}=130.29\\&\theta=arctan(\frac{40}{124})=17.88^{\circ}\\&z=-121\end{align*}$

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Example Question #143 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-67,14,138).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(68.45,168.2^{\circ},138)$

$(68.45,11.8^{\circ},138)$

$(53,11.8^{\circ},138)$

$(53,168.2^{\circ},138)$

Correct answer:

$(68.45,168.2^{\circ},138)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-67,14,138)\\&r=\sqrt{(-67)^2+(14)^2}=68.45\\&\theta=arctan(\frac{14}{-67})=168.2^{\circ}\\&z=138\end{align*}$

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Example Question #144 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (140,-103,142).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(37,-36.34^{\circ},142)$

$(37,143.66^{\circ},142)$

$(173.81,143.66^{\circ},142)$

$(173.81,-36.34^{\circ},142)$

Correct answer:

$(173.81,-36.34^{\circ},142)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(140,-103,142)\\&r=\sqrt{(140)^2+(-103)^2}=173.81\\&\theta=arctan(\frac{-103}{140})=-36.34^{\circ}\\&z=142\end{align*}$

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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 #131 : Cylindrical Coordinates

Example Question #222 : 3 Dimensional Space

Example Question #131 : Cylindrical Coordinates

Example Question #224 : 3 Dimensional Space

Example Question #132 : Cylindrical Coordinates

Example Question #133 : Cylindrical Coordinates

Example Question #141 : Cylindrical Coordinates

Example Question #142 : Cylindrical Coordinates

Example Question #143 : Cylindrical Coordinates

Example Question #144 : Cylindrical Coordinates

All Calculus 3 Resources

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