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Cartesian, spherical polar and cylindrical coordinates

Vector analysis and coordinate systems · Physics

Study notes

Convert the point (x, y, z) = (1, 1, 1) to spherical polar coordinates (r, theta, phi). Step 1: Calculate r (distance from origin). r = sqrt(x^2 + y^2 + z^2) = sqrt(1+1+1) = sqrt(3). Step 2: Calculate theta (angle from positive z-axis). cos(theta) = z/r = 1/sqrt(3). So, theta = arccos(1/sqrt(3)). Step 3: Calculate phi (angle in xy-plane from positive x-axis). tan(phi) = y/x = 1/1 = 1. Since x and y are positive, phi is in the first quadrant. phi = arctan(1) = pi/4. Final Answer: (sqrt(3), arccos(1/sqrt(3)), pi/4).

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