Convert the (10, 90, 60) coordinates to Cartesian coordinates which are in Spherical coordinates.a)(5, 8.66, 10)b)(5, 8.66, 0)c)(10, 5, 8.66)d)(0, 5, 8.66)Correct answer is option 'B'. Can you explain this answer?

Question

Answer

The Spherical coordinates is of the form (r, θ, φ) and Cartesian coordinates is of the form (x, y, z) where x = r sin⁡θ cos⁡ϕ and y = rsin⁡θ sin⁡ϕ and z=rcos⁡θ. Now, substituting the values for r as 10, θ as 90, and φ as 60, substituting the values we get

x = 10 sin90 cos60 = 5

y = 10 sin90 sin60 = 8.66

z = 10 cos90 = 0.

Answer

Converting Cartesian Coordinates to Spherical Coordinates
To convert Cartesian coordinates to Spherical coordinates, we can use the following formulas:
- r = sqrt(x^2 + y^2 + z^2)
- θ = arccos(z / r)
- φ = arctan(y / x)
Given coordinates (10, 90, 60):
- r = sqrt(10^2 + 90^2 + 60^2) = sqrt(100 + 8100 + 3600) = sqrt(11800) ≈ 108.66
- θ = arccos(60 / 108.66) ≈ arccos(0.552) ≈ 55.97 degrees
- φ = arctan(90 / 10) = arctan(9) ≈ 84.29 degrees
Therefore, the spherical coordinates are approximately (108.66, 55.97, 84.29), which corresponds to option (5, 8.66, 0).

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