Answered: How do you express the coordinates of…

Principles of Physics: A Calculus-Based Text

5th Edition

ISBN:9781133104261

Author:Raymond A. Serway, John W. Jewett

Publisher:Raymond A. Serway, John W. Jewett

Chapter1: Introduction And Vectors

Section: Chapter Questions

Problem 30P: Two points in the xy plane have Cartesian coordinates (2.00, 4.00) m and (3.00, 3.00) m. Determine...

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How do you express the coordinates of an object in cylindrical and spherical coordinates when its coordinates in a Cartesian reference frame are (3, 4, 5). Assume that all three coordinate systems share a common origin.

Expert Solution

Concept:

The Cartesian coordinates $(x, y, z)$ of a point is related to cylindrical coordinate $(r, θ, z)$ as, $x = r \cos (θ), y = r \sin (θ), z = z$ . For conversion from Cartesian to cylindrical we can use the following relations: $r = \sqrt{x^{2} + y^{2}}, θ = \tan^{- 1} (\frac{y}{x}), z = z$ .

The Cartesian coordinates $(x, y, z)$ of a point is related to spherical coordinate $(ρ, θ, φ)$ as, $x = ρ \sin (φ) \cos (θ), y = ρ \sin (φ) \sin (θ), z = ρ \cos (φ)$ . For conversion from Cartesian to cylindrical we can use the following relations: $ρ = \sqrt{x^{2} + y^{2} + z^{2}}, θ = \tan^{- 1} (\frac{y}{x}), φ = \cos^{- 1} (\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}})$ .

Given: $(x, y, z) = (3, 4, 5)$

Required: $(r, θ, z)$ and $(ρ, θ, φ)$ .

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