In this question you have to find the Spherical Coordinates ( , theta , phi )of

Question

In this question you have to find the Spherical Coordinates ( , theta , phi )of a point given in Cartesian Coordinates (x,y,z). The correct answer will be one of the following: Hint: Given the above, considering just the signs of x, y, z, or whether they are zero, is enough to determine the correct theta and phi . If (x,y,z) = (1,1, -Sqrt(2)), then ( , theta , phi ) = Answer is a of the form (1 ,pi/2,pi/3) If (x,y,z) = (-Sqrt(2), Sqrt(2), 0), then ( ,theta , phi ) = Answer is a of the form (1 ,pi/2,pi/3) If (x,y,z) = (Sqrt(2), -Sqrt(2), 0), then ( , theta , phi ) = Answer is a of the form (1,pi/2,pi/3)

Explanation / Answer

1. (x,y,z) = (1,1,-sqrt 2)

spherical co-ordinates (rho, theta, phi)

rho = sqrt of (x^2 +y^2 +z^2) = sqrt ( 1 + 1 + 2 ) = sqrt (4) = 2

phi = arc cos ( z/rho) = arc cos (sqrt 2 /2) = arc cos (1/sqrt 2) = pi/4

theta = arc sin (y/sqrt (x^2 +y^2)) = arc sin (1/sqrt2) = pi/4

2. rho = sqrt ( 2+2) = 2

phi = arc cos (0/rho) = pi/2

theta = arc sin (sqrt 2 / 2 ) = pi/4

3. rho = 2

phi = pi/2

theta = arc sin (-sqrt 2/2) = arc sin (-1/sqrt 2) = pi-pi/4 = 3pi/4

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