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

$(ρ, θ, ϕ) = (6 \sqrt{3}, \frac{π}{4}, \cos^{- 1} (\frac{1}{\sqrt{3}}))$

$(r, θ, ϕ) = (6 \sqrt{2}, - \frac{π}{4}, 6)$

Step by step solution

Given Information

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

Step 2:Finding the Cylindrical coordinates

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

We know that

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

Using given values, we get

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

$\Rightarrow r = \sqrt{{(- 6)}^{2} + 6^{2}}, θ = \tan^{- 1} - 1, z = 6$

$r = 6 \sqrt{2}, θ = - \frac{π}{4}, z = 6$

localid="1652247754695" $\Rightarrow (r, θ, ϕ) = (6 \sqrt{2}, - \frac{π}{4}, 6)$

Finding the spherical coordinates

We know that

ρ = \sqrt{x^{2} + y^{2} + z^{2}}, θ = \tan^{- 1} \frac{y}{x}, ϕ = \cos^{- 1} \frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}

Using values

role="math" localid="1652247891309" $ρ = \sqrt{{(- 6)}^{2} + {(6)}^{2} + {(6)}^{2}}, θ = \tan^{- 1} \frac{6}{- 6}, ϕ = \cos^{- 1} \frac{6}{\sqrt{{(- 6)}^{2} + {(6)}^{2} + {(6)}^{2}}}$

$\Rightarrow ρ = \sqrt{36 + 36 + 36}, θ = \tan^{- 1} - 1, ϕ = \cos^{- 1} \frac{6}{\sqrt{36 + 36 + 36}}$

$\Rightarrow ρ = 6 \sqrt{3}, θ = \frac{π}{4}, ϕ = \cos^{- 1} (\frac{1}{\sqrt{3}})$

$(ρ, θ, ϕ) = (6 \sqrt{3}, \frac{π}{4}, \cos^{- 1} (\frac{1}{\sqrt{3}}))$

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

Short Answer

Step by step solution

Given Information

Step 2:Finding the Cylindrical coordinates

Finding the spherical coordinates

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

Short Answer

Step by step solution

Given Information

Step 2:Finding the Cylindrical coordinates

Finding the spherical coordinates

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Pure Maths

Discrete Mathematics

Statistics

Probability and Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

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