Question

Practice: 1. A vector defined at (4,0,0) is expressed in Cartesian coordinate system as (2,3,0), convert it into cylindrical and spherical representations; 2. If the same vector is moved to (4,0,1), what are its cylindrical and spherical representations?

Answer 1

Convergesion of Cartesian coordinates into Cylindrical cordinale, pus litro un all The cartesian coordinate (X,Y,Z) and the cylindrical coordinates (rico, z) of a point are related as follow : 5 - These equations are wed to convert from rectangular Coordinate to cylindrical coordinale accarlesian) 1. 22 = x2 + y2 01/2. tanco = 91x ut, oso sal vivaro 132 ZZ O se since we have cartesian Cordinate Cary, z) = (2,3,0) liwe convert it into cylindrical coordinate.. 72x2+y? 22+32 24+9 r2 = 13 fat 13 il we choose the positive square root of How to find annab wirool nt wond In this case both x and y are positive a lies between 0 to 12 tancy - 3 22 a = tant (3/2) - 56.310 and z= o . Cylindrical coordinales are. (T13, 56.31,0)

Conversion of cartesian coordinak to spherical coordinate These equation are wed to convert from cartesian coordinate (x, y, 2) to spherical coordinale (904) PP22+ y2 +22 tano = 4122 Y = arc cos Y T 182773722 Here (aly ,2)=(2,3,0) p² = x2 + y2 + 2² - 2 ² + 3²2 9 anton = 4+9+0 - 13 : : 82- 13 § - I 113 we choose positive Square root, . = 1135 tano = 1/x - 3 @ = tant ( 312) 256.310 and V COS cos " x2 + y2 +2² Y = cos(0) Jutg to Hence the spherical coordinates are c.0.) = ( J13, 56.31 , go)

part 1 representation of using the same argument we use in we find cylindrical and spherical exy, 2) 2 (401) a) Cylindrical representation - 2. x² + y² = 42to2 = 16 72= 16 rat 16 we choose positive square rool, tano ano o and i tast(o)=0 = 1 cylindrical coordinates are (8, 0, 2) = (4,0, 1) 6) spherical representation - p² = x² + y2 + z² = 4²40 24 12 1. P2 = 17 = 16+1 tano = 1/x = 0 .0 = lani (o) = 0 COS Ja2ty2 +22 9 - HT. i spherical coordinates are (s, 0,4)= ($17, 0,75.96).