Find a Cartesian equation for the set S of all points 6 units above the xy-plane

Question

Find a Cartesian equation for the set S of all points 6 units above the xy-plane. Now find cylindrical and spherical equations for S. Find a Cartesian equation for the set S of all points 3 units to the left of the yz-plane. Now find cylindrical and spherical equations for S. Find a cylindrical equation for the set of all points in R^3 which are 4 units from the z-axis. Find a spherical equation for the set S of all points 5 units from the origin. Now find Cartesian and cylindrical equations for 5. Find a Cartesian equation for the sphere centered at (0, 0, 1) with radius 1. Now find a polar equation for it. Describe, as carefully as you can, the intersection of the constant coordinate surfaces given below. Include what geometric shape it is (e.g. a line, ray, circle, etc.), and how it sits in R^3 (e.g. horizontally, parallel to the y-axis, etc

Explanation / Answer

13. Cartesian equation of a plane i.e. 6 units of above xy plane is z = 6, with coordinates (x,y,6)

Cylindrical equation of a plane i.e. 6 units of above xy plane is z = 6, with coordinates (,,6)

where = (x2 + y2), = tan-1(y/x)

Sphherical equation of a plane i.e. 6 units of above xy plane is z = 6, with coordinates (r,,)

where r = (x2 + y2 + 62), = tan-1(y/x), = tan-1((x2 + y2)/z) = tan-1((x2 + y2)/6)

15. Let the point be (x,y,z) then its distance from (0,0,z) is 4 units.

=> ((x-0)2 + (y-0)2 + (z-z)2) = (x2 + y2) = 42

=> x2 + y2 = 16

So cylindrical equation is 2 = 16

=> = 4 or = -4

with coordinates (,,z)

where = (x2 + y2), = tan-1(y/x)

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