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Calculus 3 : 3-Dimensional Space

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6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #245 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (113,2^{\circ},-74).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(112.93,3.94,-74)

(3.94,112.93,-74)

(102.75,-47.02,-74)

(-47.02,102.75,-74)

Correct answer:

(112.93,3.94,-74)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=113cos(2^{\circ})=112.93\\&y=113sin(2^{\circ})=3.94\\&z=-74\end{align*}$

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Example Question #1991 : Calculus 3

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (105,166^{\circ},118).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-101.88,25.4,118)

(25.4,-101.88,118)

(-91.92,50.75,118)

(50.75,-91.92,118)

Correct answer:

(-101.88,25.4,118)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=105cos(166^{\circ})=-101.88\\&y=105sin(166^{\circ})=25.4\\&z=118\end{align*}$

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Example Question #1992 : Calculus 3

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (83,-127^{\circ},-109).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-49.95,-66.29,-109)

(-80.73,19.29,-109)

(-66.29,-49.95,-109)

(19.29,-80.73,-109)

Correct answer:

(-49.95,-66.29,-109)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=83cos(-127^{\circ})=-49.95\\&y=83sin(-127^{\circ})=-66.29\\&z=-109\end{align*}$

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Example Question #248 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (39,-89^{\circ},103).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(0.68,-38.99,103)

(-33.54,19.9,103)

(19.9,-33.54,103)

(-38.99,0.68,103)

Correct answer:

(0.68,-38.99,103)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=39cos(-89^{\circ})=0.68\\&y=39sin(-89^{\circ})=-38.99\\&z=103\end{align*}$

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Example Question #331 : 3 Dimensional Space

Find a parametric representation of the circle 3x^2 + 9 y ^2 = 27 .

Possible Answers:

$\vec{r}(t)= \left< \cos 3t, \sqrt{3} \sin 3t \right>$

$\vec{r}(t)= \left< 3 \cos t, \sqrt{3} \sin t \right>$

$\vec{r}(t)= \left< 3 \cos t, \sqrt{3} \tan t \right>$

$\vec{r}(t)= \left< \cos t, \sin t \right>$

Correct answer:

$\vec{r}(t)= \left< 3 \cos t, \sqrt{3} \sin t \right>$

Explanation:

We can begin by rewriting the equation for a circle as

$3x^2 + 9y^2 = 27 \Rightarrow \left(\frac{x}{3} \right )^2 + \left(\frac{y}{\sqrt{3}} \right )^2=1$ .

This directly tells us that $x = 3 \cos t, y = \sqrt{3} \sin t$ . This allows us to write our final expression for the parametric representation as

$\vec{r}(t)= \left< 3 \cos t, \sqrt{3} \sin t \right>$

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Example Question #242 : Cylindrical Coordinates

Convert the following vector in Cartesian coordinates into cylindrical coordinates.

(3xy, x^2z, y)

Possible Answers:

$(3r\cos\theta \sin \theta, rz\cos^2 \theta, r\sin \theta )$

$(3r^2\sin\theta \cos\theta, rz\sin^2 \theta, r\cos \theta )$

$(3r^2\cos\theta \sin \theta, rz\cos^2 \theta, r\sin \theta )$

$(3r^2\sin\theta \cos\theta, r\sin^2 \theta, \cos \theta )$

$(3r^2\sin\theta \cos\theta, r\sin^2 \theta, r\cos \theta )$

Correct answer:

$(3r^2\cos\theta \sin \theta, rz\cos^2 \theta, r\sin \theta )$

Explanation:

The conversion from Cartesian to cylindrical coordinates is as follows:

$x=r\cos \theta$

$y=r \sin \theta$

z=z

The three components of the vector then become:

$3xy\rightarrow3r\cos \theta \:r\sin \theta=3r^2\cos\theta \sin\theta$

$x^2z\rightarrow r^2 \cos^2\theta z$

$y\rightarrow r\sin \theta$

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Example Question #251 : Cylindrical Coordinates

Express the three-dimensional (x,y,z) Cartesian coordinates as cylindrical coordinates (r, θ, z):

$\left ( 2,1, -2\right )$

Possible Answers:

$\left ( \sqrt{5}, \tan^{-1}2, -2 \right )$

$\left ( \sqrt{5}, \tan^{-1}\frac{1}{2}, -2 \right )$

$\left ( 5, \tan^{-1}\frac{1}{2}, -2 \right )$

$\left ( 5, \tan^{-1}2, -2 \right )$

$\left ( \sqrt{5}, \tan^{-1}\frac{1}{2}, 1 \right )$

Correct answer:

$\left ( \sqrt{5}, \tan^{-1}\frac{1}{2}, -2 \right )$

Explanation:

The coordinates (2, 1, -2) corresponds to: x = 2, y = 1, z = -2, and are to be converted to the cylindrical coordinates in form of (r, θ, z), where:

$\\ r = \sqrt{x^{2}+y^{2}} \\ \tan(\theta) = \frac{y}{x} \\ z = z$

So, filling in for x, y, z:

$\\ r = {\sqrt{2^{2}+1^{2}}} = \sqrt{5} \\ \tan(\theta) = \frac{1}{2} \Rightarrow \theta = \tan^{-1}\frac{1}{2} \\ z = -2$

Then the cylindrical coordinates are represented as:

$\left ( \sqrt{5}, \tan^{-1}\frac{1}{2}, -2 \right )$

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Example Question #252 : Cylindrical Coordinates

Express the three-dimensional (x,y,z) Cartesian coordinates as cylindrical coordinates (r, θ, z):

$\left (0, 3, 4\right )$

Possible Answers:

$\left (9, \frac{3}{0}, 4\right )$

$\left (3, \frac{3}{0}, 4\right )$

$\left (3, \frac{1}{2}\pi, 4\right )$

$\left (3, \pi, 4\right )$

$\left (9, \frac{1}{2}\pi, 4\right )$

Correct answer:

$\left (3, \frac{1}{2}\pi, 4\right )$

Explanation:

The coordinates (0, 3, 4) corresponds to: x = 0, y = 3, z = 4, and are to be converted to the cylindrical coordinates in form of (r, θ, z), where:

$\\ r = \sqrt{x^{2}+y^{2}} \\ \tan(\theta) = \frac{y}{x} \\ z = z$

So, filling in for x, y, z:

$\\ r = {\sqrt{0^{2}+3^{2}}} = \sqrt{9} = 3 \\ \tan(\theta) = \frac{3}{0} \Rightarrow \theta = \frac{1}{2}\pi \\ z = 4$

Then the cylindrical coordinates are represented as:

$\left (3, \frac{1}{2}\pi, 4 \right )$

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Example Question #253 : Cylindrical Coordinates

Express the three-dimensional (x,y,z) Cartesian coordinates as cylindrical coordinates (r, θ, z):

$\left (\sqrt{2}, 1, 1\right )$

Possible Answers:

$\left (9, \tan^{-1}\frac{\sqrt{2}}{2}, 1\right )$

$\left (3, \frac{\sqrt{2}}{2}, 1\right )$

$\left ( \sqrt{3}, \tan^{-1}\frac{\sqrt{2}}{2}, 1\right )$

$\left ( \sqrt{3}, \frac{\sqrt{2}}{2}, 1\right )$

$\left (3, \tan^{-1}\frac{\sqrt{2}}{2}, \sqrt{2}\right )$

Correct answer:

$\left ( \sqrt{3}, \tan^{-1}\frac{\sqrt{2}}{2}, 1\right )$

Explanation:

The coordinates (√2, 1, 1) corresponds to: x = √2, y = 1, z = 1, and are to be converted to the cylindrical coordinates in form of (r, θ, z), where:

$\\ r = \sqrt{x^{2}+y^{2}} \\ \tan(\theta) = \frac{y}{x} \\ z = z$

So, filling in for x, y, z:

$\\ r = {\sqrt{\sqrt{2}^{2}+1^{2}}} = \sqrt{2+1} = \sqrt{3} \\ \tan(\theta) = \frac{1}{\sqrt{2}} \Rightarrow \theta = \tan^{-1}\frac{\sqrt{2}}{2} \\ z = 1$

Then the cylindrical coordinates are represented as:

$\left (\sqrt{3}, \tan^{-1}\frac{\sqrt{2}}{2}, 1 \right )$

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Example Question #254 : Cylindrical Coordinates

Express the three-dimensional cylindrical coordinates (r, θ, z) as three-dimensional (x,y,z) Cartesian coordinates:

$\left (3, \frac{1}{3}\pi, -4 \right)$

Possible Answers:

$\left (\frac{3}{2}, \frac{3\sqrt{3}}{2}, -4 \right)$

$\left (3\cos^{-1}\frac{1}{2}, 3\sin^{-1}\frac{\sqrt{3}}{2}, -4 \right)$

$\left (\frac{1}{2}, \frac{\sqrt{3}}{2}, -4 \right)$

$\left (3\pi, 3\pi, -4 \right)$

$\left (\frac{3}{2}, \frac{\sqrt{3}}{2}, -4 \right)$

Correct answer:

$\left (\frac{3}{2}, \frac{3\sqrt{3}}{2}, -4 \right)$

Explanation:

The coordinates (3, π/3, -4) corresponds to: r = 3, θ = π/3, z = -4, and are to be converted to the Cartesian coordinates in form of (x, y, z), where:

$\\ x = r\cos\theta \\ y = r\sin\theta \\ z = z$

So, filling in for r, θ, z:

$\\ x = 3\cos{\frac{1}{3}\pi} = 3\cdot \frac{1}{2} = \frac{3}{2} \\ y = 3\sin{\frac{1}{3}\pi} = 3\cdot \frac{\sqrt{3}}{2} = \frac{3\sqrt{2}}{2} \\ z = -4$

Then the Cartesian coordinates are represented as:

$\left (\frac{3}{2}, \frac{3\sqrt{3}}{2}, -4 \right)$

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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 #245 : Cylindrical Coordinates

Example Question #1991 : Calculus 3

Example Question #1992 : Calculus 3

Example Question #248 : Cylindrical Coordinates

Example Question #331 : 3 Dimensional Space

Example Question #242 : Cylindrical Coordinates

Example Question #251 : Cylindrical Coordinates

Example Question #252 : Cylindrical Coordinates

Example Question #253 : Cylindrical Coordinates

Example Question #254 : Cylindrical Coordinates

All Calculus 3 Resources

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