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Chapter 13: Q 24. (page 1066)

Give the cylindrical and spherical coordinates for the point with rectangular coordinates $(- 6, 6, 6)$

Short Answer

Expert verified

$(蚁, 胃, 蠒) = (6 \sqrt{3}, \frac{蟺}{4}, \cos^{- 1} (\frac{1}{\sqrt{3}}))$

$(r, 胃, 蠒) = (6 \sqrt{2}, - \frac{蟺}{4}, 6)$

Step by step solution

01

Given Information

Given data is $x = - 6, y = 6, z = 6$

02

Step 2:Finding the Cylindrical coordinates

It is given that $x = - 6, y = 6, z = 6$

We know that

$r = \sqrt{x^{2} + y^{2}}, 胃 = \tan^{- 1} \frac{y}{x}, z = z$

Using given values, we get

localid="1652247726196" $r = \sqrt{{(- 6)}^{2} + 6^{2}}, 胃 = \tan^{- 1} \frac{6}{- 6}, z = 6$

$鈬� r = \sqrt{{(- 6)}^{2} + 6^{2}}, 胃 = \tan^{- 1} - 1, z = 6$

$r = 6 \sqrt{2}, 胃 = - \frac{蟺}{4}, z = 6$

localid="1652247754695" $鈬� (r, 胃, 蠒) = (6 \sqrt{2}, - \frac{蟺}{4}, 6)$

03

Finding the spherical coordinates

We know that

蚁 = \sqrt{x^{2} + y^{2} + z^{2}}, 胃 = \tan^{- 1} \frac{y}{x}, 蠒 = \cos^{- 1} \frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}

Using values

role="math" localid="1652247891309" $蚁 = \sqrt{{(- 6)}^{2} + {(6)}^{2} + {(6)}^{2}}, 胃 = \tan^{- 1} \frac{6}{- 6}, 蠒 = \cos^{- 1} \frac{6}{\sqrt{{(- 6)}^{2} + {(6)}^{2} + {(6)}^{2}}}$

$鈬� 蚁 = \sqrt{36 + 36 + 36}, 胃 = \tan^{- 1} - 1, 蠒 = \cos^{- 1} \frac{6}{\sqrt{36 + 36 + 36}}$

$鈬� 蚁 = 6 \sqrt{3}, 胃 = \frac{蟺}{4}, 蠒 = \cos^{- 1} (\frac{1}{\sqrt{3}})$

$(蚁, 胃, 蠒) = (6 \sqrt{3}, \frac{蟺}{4}, \cos^{- 1} (\frac{1}{\sqrt{3}}))$

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