Question 2:Three frequently used coordinates in R³ are 1: Cartesian Coordinates, 2: Cylindrical coordi-nates, and 3: Spherical Coordinates. Unit vectors in these coordinates are the following.Cartesian coordinates (r, y, z):êz = î,êy = 3,ê̟ = k.(1)(2)(3)Cylindrical coordinates (p, ø, z):ê, = cos o î + sin ø j,ês = - sin o i + cos o 3,êz = k(4)(5)(6)Spherical coordinates (r, 0, ø):ê, = sin 0 cos o î+ sin 0 sin ø ĵ + cos 0 k,êg = cos 0 cos o i+ cos 0 sin ø j – sin 0 k,ês = - sin o î + cos o j.(7)(8)COS(9)a) Find the expressions of (êr, êo, ês) in terms of (êp, ês, êz).b) A vector can be expressed in terms of any basis. In particular, we can write:V = V,ê, + V,êy + V,ê; = V,ê, + Vgêg + Vệê; = V,ê, + Vạêp + Viêg.(10)In terms of column vector notation we write(VeV,), Vci = [ V), Vsp(11)Vcr =V:VeThe relations between them can be expressed in terms of matrix:Vcr = Pcr+-ciVcı, Vcr = Pcr+SpVsp;Vci = Pcu-CrVor, Vci = Pcl-SpVsp:Vsp = Pspt-Cr Vcr; Vsp = Psp+-CiVci,(12)(13)(14)where Pcrt-Cl, Pcre-Sp; Pcu-Cr; Pcu-sSp; Pspt-Cr; Pspt-Cl are all 3x3 matrices. Without explicitlyfinding the matrices show that:Psp-CrPcre-Sp = I3, Psp-Cl = Pspe-CPC+-Cl-(15)%3D%3Dc) Now explicitly compute Pspe-Cl and show that Pspe-Cl E SO(3).

Name: Question 2:Three frequently used coordinates in R³ are 1: Cartesian C
Uploaded: 2024-03-20T06:46:42.000Z
Channel: Bartleby
Description: Question 2:Three frequently used coordinates in R³ are 1: Cartesian Coordinates, 2: Cylindrical coordi-nates, and 3: Spherical Coordinates. Unit vectors in these coordinates are the following.Cartesian coordinates (r, y, z):êz = î,êy = 3,ê̟ = k.(1)(