Convert the point given from Cartesian coordinates into Sphe

Convert the point given from Cartesian coordinates into Spherical coordinates and Cylindrical coordinates. Use the squareroot symbol \'squareroot\' where needed to give an exact value for your answer. Be sure to include parentheses where necessary, e.g. to distinguish 1/(2k) from 1/2k. Cartesian: (-squareroot 3/squareroot 2, 1/squarerot 2, - squareroot 2) rightarrow Spherical: or Cylindrical:

Solution

We know that the spherical coordinates corresponding to the cartesian coordinates (x,y,z) are (,,Ø), where = (x² + y² +z²) , = tan^-1(y/x) and Ø = tan^-1[(x² +y²)/z]. Here x = -3/2 , y = 1/2 and z = -2. Therefore, = (3/2+1/2+2)=4 =2.Also, =tan^-1(-1/3) = 60⁰ = /3. Also, Ø = tan^-1( 2/-2) = tan^-1(-2)= -54.74⁰ = 0.3. Thus, the spherical coordinates corresponding to the given cartesian coordinates are (2,/3,0.3).

We know that the spherical coordinates corresponding to the cartesian coordinates (x,y,z) are (r,,z), where r = (x² + y²) and = tan^-1(y/x). Then r = 2 Further (as above) = /3. Hence, the cylindrical coordinates corresponding to the given cartesian coordinates are ( 2, /3,-2)