Cylindrical Coordinates

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AP Calculus BC › Cylindrical Coordinates

Questions 1 - 10

A point in space is located, in Cartesian coordinates, at . What is the position of this point in cylindrical coordinates?

$(9.43,147.99^{\circ},0)$

$(9.43,32.01^{\circ},0)$

$(11.23,-32.01^{\circ},0)$

$(11.23,32.01^{\circ},0)$

$(9.43,-32.01^{\circ},0)$

Explanation

When given Cartesian coordinates of the form to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

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

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

$r=\sqrt{x^2+y^2}\rightarrow9.43$

$\theta=arctan(\frac{y}{x})\rightarrow 147.99^{\circ}$ (Bearing in mind sign convention)

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

Explanation

When given cylindrical coordinates of the form $(r,\theta,z)$ and converting to Cartesian coordinates of the form , the relationship between , , , and $\theta$ will be of use:

$x=rcos(\theta)$

$y=rsin(\theta)$

(This value is identical across Cartesian and cylindrical coordinate systems)

For the cylindrical coordinates

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

$x=rcos(\theta)\rightarrow0$

$y=rsin(\theta)\rightarrow5$

$z=z\rightarrow7$

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

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

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

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

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

Explanation

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

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (59,-55,136).\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

$(80.66,-42.99^{\circ},136)$

$(80.66,137.01^{\circ},136)$

$(4,-42.99^{\circ},136)$

$(4,137.01^{\circ},136)$

Explanation

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\& \text{are the most straightforward: }\& r=\sqrt{x^2+y^2}\&z=z\&\text{However, it is prudent to be carful when calculating }\theta.\&\text{The formula for it is as follows:}\& \theta=arctan(\frac{y}{x})\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\& \text{which quadrant the point lies in; this determines the value of }\theta.\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\&\text{values lead to }|\theta|>90^{\circ}\&\&\text{For our coordinates: }(59,-55,136)\&r=\sqrt{(59)^2+(-55)^2}=80.66\&\theta=arctan(\frac{-55}{59})=-42.99^{\circ}\&z=136\end{align*}$

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (53,-90^{\circ},-91).\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Explanation

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\&x=rcos(\theta)\&y=rsin(\theta)\&\text{Finding }z\text{ is much simpler; it does not change between}\& \text{Cartesian and cylindrical coordinates:}\&z=z\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\& \text{imperative that the correct units (degrees or radians) are specified}\& \text{for the input!}\&\text{For our coordinates}\&x=53cos(-90^{\circ})=0\&y=53sin(-90^{\circ})=-53\&z=-91\end{align*}$

A point in space is located, in Cartesian coordinates, at . What is the position of this point in cylindrical coordinates?

$(33.84,-18.97^{\circ},37)$

$(33.84,161.03^{\circ},37)$

$(21,161.03^{\circ},37)$

$(21,-18.97^{\circ},37)$

Explanation

When given Cartesian coordinates of the form to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

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

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

$\begin{align*}&r=\sqrt{(32)^2+(-11)^2}=33.84\&\theta=arctan(\frac{-11}{32})=-18.97^{\circ}\&z=37\end{align*}$

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (9,101^{\circ},9).\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Explanation

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\&x=rcos(\theta)\&y=rsin(\theta)\&\text{Finding }z\text{ is much simpler; it does not change between}\& \text{Cartesian and cylindrical coordinates:}\&z=z\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\& \text{imperative that the correct units (degrees or radians) are specified}\& \text{for the input!}\&\text{For our coordinates}\&x=9cos(101^{\circ})=-1.72\&y=9sin(101^{\circ})=8.83\&z=9\end{align*}$

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (75,-58^{\circ},138).\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Explanation

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\&x=rcos(\theta)\&y=rsin(\theta)\&\text{Finding }z\text{ is much simpler; it does not change between}\& \text{Cartesian and cylindrical coordinates:}\&z=z\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\& \text{imperative that the correct units (degrees or radians) are specified}\& \text{for the input!}\&\text{For our coordinates}\&x=75cos(-58^{\circ})=39.74\&y=75sin(-58^{\circ})=-63.6\&z=138\end{align*}$

A point in space is located, in Cartesian coordinates, at . What is the position of this point in cylindrical coordinates?

$(13,-22.62^{\circ},31)$

$(13,22.62^{\circ},31)$

$(13,157.38^{\circ},31)$

$(7,-22.62^{\circ},31)$

$(7,157.38^{\circ},31)$

Explanation

When given Cartesian coordinates of the form to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

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

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

$\begin{align*}&r=\sqrt{(12)^2+(-5)^2}=13\&\theta=arctan(\frac{-5}{12})=-22.62^{\circ}\&z=31\end{align*}$

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (107,-81^{\circ},77).\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Explanation

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\&x=rcos(\theta)\&y=rsin(\theta)\&\text{Finding }z\text{ is much simpler; it does not change between}\& \text{Cartesian and cylindrical coordinates:}\&z=z\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\& \text{imperative that the correct units (degrees or radians) are specified}\& \text{for the input!}\&\text{For our coordinates}\&x=107cos(-81^{\circ})=16.74\&y=107sin(-81^{\circ})=-105.68\&z=77\end{align*}$

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