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To convert the given rectangular coordinates (A(2,3,1)) into the corresponding cylindrical coordinates, we need to use the conversion formulas from Cartesian (rectangular) coordinates ((x,y,z)) to cylindrical coordinates ((r,theta,z)):
1. (r = sqrt{x^2 + y^2})
2. (theta = tan^{-1}(frac{y}{x}))
3. (z = z)
Given (A(2,3,1)), where (x=2), (y=3), and (z=1), let’s apply these formulas:
1. (r = sqrt{2^2 + 3^2} = sqrt{4 + 9} = sqrt{13})
2. (theta = tan^{-1}(frac{3}{2})) (To get the angle in radians or degrees, remember that (tan^{-1}(frac{3}{2})) is approximately (0.9828) radians or about (56.31^circ), but the actual value will depend on the specific trigonometric function used by your calculator or software, considering the quadrant in which the point lies.)
3. (z = 1)
So, the cylindrical coordinates of (A(2,3,1)) are ((sqrt{13}, 0.9828
d
Explanation: ρ = √(x2+y2
) = √13 = 3.61
Φ = tan-1
(y/x) = 56.31
z = 1
Thus, A = (3.61,56.31,1).