Chapter 13: Q 28. (page 1066)

Give the rectangular and cylindrical coordinates for the point with spherical coordinates:
$(6, \frac{π}{4}, \frac{π}{4})$

Short Answer

Expert verified

$(x, y, z) = (3, 3, 3 \sqrt{2})$

$(r, θ, z) = (3 \sqrt{2}, \frac{π}{4}, 3 \sqrt{2})$

Step by step solution

Given Information

The given expression is $(ρ, θ, ϕ) = (6, \frac{π}{4}, \frac{π}{4})$

To find rectangular coordinates

As $ρ = 6, θ = \frac{π}{4}, ϕ = \frac{π}{4}$

We know that

$x = ρ \sin ϕ \cos θ, y = ρ \sin ϕ \sin θ, z = ρ \cos ϕ$

Substituting values, we get

$x = 6 \sin \frac{π}{4} \cos \frac{π}{4}, y = 6 \sin \frac{π}{4} \sin \frac{π}{4}, z = 6 \cos \frac{π}{4}$

$\Rightarrow x = 6 \frac{1}{\sqrt{2}} \frac{1}{\sqrt{2}}, y = 6 \frac{1}{\sqrt{2}} \frac{1}{\sqrt{2}}, z = 6 \frac{1}{\sqrt{2}}$

$x = 3, y = 3, z = 3 \sqrt{2}$

Hence, rectangular coordinates are $(x, y, z) = (3, 3, 3 \sqrt{2})$

To find cylindrical coordinates

To find cylindrical coordinates from spherical coordinates, we use

$r = ρ \sin ϕ, θ = θ, z = ρ \cos ϕ$

Using given values of spherical coordinates, we get

$r = 6 \sin \frac{π}{4}, θ = \frac{π}{4}, z = 6 \cos \frac{π}{4}$

$r = 6 \frac{1}{\sqrt{2}}, θ = \frac{π}{4}, z = 6 \frac{1}{\sqrt{2}}$

$\Rightarrow r = 3 \sqrt{2}, θ = \frac{π}{4}, z = 3 \sqrt{2}$

Hence, cylindrical coordinates are $(r, θ, z) = (3 \sqrt{2}, \frac{π}{4}, 3 \sqrt{2})$

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Give the rectangular and cylindrical coordinates for the point with spherical coordinates:
$(6, \frac{π}{4}, \frac{π}{4})$

Short Answer

Step by step solution

Given Information

To find rectangular coordinates

To find cylindrical coordinates

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Most popular questions from this chapter

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Give the rectangular and cylindrical coordinates for the point with spherical coordinates:
$(6, \frac{π}{4}, \frac{π}{4})$