Chapter 13: Q 26. (page 1066)

Give the rectangular and spherical coordinates for the point with cylindrical coordinates $(4, \frac{π}{3}, 6)$

Short Answer

Expert verified

$(x, y, z) = (2, 2 \sqrt{3}, 6)$

$(ρ, θ, ϕ) = ({\sqrt{52}, \frac{π}{3}, \tan}^{- 1} \frac{2}{63})$

Step by step solution

Given Information

The given expression is $(r, θ, z) = (4, \frac{π}{3}, 6)$

To find rectangular coordinates

As $r = 4, θ = \frac{π}{3}, z = 6$

We know that for converting cylindrical to rectangular coordinates, we use

$x = r \cos θ, y = r \sin θ, z = z$

localid="1652288468915" $x = 4 \cos \frac{π}{3}, y = 4 \sin \frac{π}{3}, z = 6$

$x = 4 \frac{1}{2}, y = 4 \frac{\sqrt{3}}{2}, z = 6$

$x = 2, y = 2 \sqrt{3}, z = 6$

Hence, $(x, y, z) = (2, 2 \sqrt{3}, 6)$

To find spherical coordinates

For converting cylindrical to spherical coordinates, we use

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

Using given values,

$ρ = {\sqrt{4^{2} + 6^{2}}, θ = \frac{π}{3}, ϕ = \tan}^{- 1} \frac{4}{6}$

$ρ = {\sqrt{52}, θ = \frac{π}{3}, ϕ = \tan}^{- 1} \frac{2}{3}$

$\Rightarrow ρ = {2 \sqrt{13}, θ = \frac{π}{3}, ϕ = \tan}^{- 1} [\frac{2}{3}]$

Hence, $(ρ, θ, ϕ) = (2 \sqrt{13}, {\frac{π}{3}, \tan}^{- 1} [\frac{2}{3}])$

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Give the rectangular and spherical coordinates for the point with cylindrical coordinates $(4, \frac{π}{3}, 6)$

Short Answer

Step by step solution

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Give the rectangular and spherical coordinates for the point with cylindrical coordinates $(4, \frac{π}{3}, 6)$