Hello friends welcome to engineering fanda family in this video I'll explain conversion of Cartesian Vector into cylindrical Vector see in my last video I have explained cylindrical coordinate system and in that video I have explained conversion of cylindrical coordinate into cartisian coordinate and I have explained conversion of cartisian coordinate into cylindrical coordinate but in this video I'll explain conversion of cartisian vector into cylindrical Vector there is a difference in between vector and coordinate in this video I'll explain you conversion of cartisian vector into cylindrical vector and at last I'll solve one problem based on it so let us begin this session with conversion process first so first of all you need to understand what is c cian Vector let me take one example here let us consider we have Vector P that is there in form of cartisian vector so that will be ax into IX Direction plus a y into iy Direction plus a z into Iz Direction and this is what cartisian Vector that we need to convert into cylindrical Vector cylindrical parameters are R pi and Z so we are deal with to have a conversion and we need to have Vector p in form of R 5 and Z so we are deal with to have a r into I Direction plus A5 into I5 Direction plus AZ into Iz Direction so we are deal with to convert this cartisian Vector into cylindrical Vector right now I'll explain you entire process see here let us consider one point and that point that we are viewing it from Top VI so that is available somewhere over here right now with this point if You observe we have quarter cylinder over here from Top you it is appearing somewhat like this right from Top you we have XY plane you can observe now Here If You observe we have radius that radius is R we have radius that radius is r and if you take trajectory on XY plane then with respect to x-axis on XY plane this angle is pi if you take break trajectory on XY plane then with respect to xaxis this angle is five right and this Z coordinate that is length of cylinder so length of cylinder that is Z over here right now you need to understand what are the direction which is there with ir i5 and Iz and based on those Direction we can identify the value of a r A5 and a z so here what is a r direction see a r direction if I want to plot it from here then it is there in radial Direction so you see it is there in radial Direction it is there in radial Direction like this this is I Direction let me note it down now let us try to understand what is the direction of five the direction of five that is tangent to C direction of Y that is tangent to C so tangent to C from X to y direction that is direction of five so it is appearing in this direction you can observe so this is I5 direction that is tangent to C so I5 that is tangent to C from X to y direction right and see direction of Zed that is there in the direction of length of cylinder so it is appearing in this direction it is appearing in this direction right and if you carefully observe see I i5 and I Z this directions are perpendicular to each other like we have direction of x y and Zed that is perpendicular to each other similarly I i5 and I Z these three axis are perpendicular to each other right now from Top View If You observe then here we have radial direction right here we have radial direction that is I Direction and tangent to cve tangent to cve from X to Y that is in this direction that is iy Direction and I Z that is perpendicular to this page right so that will that will be dot over here so that is how directions are there now I'll explain you the process of calculation of a r A5 and AZ right so here if you want to identify a r that is radial component we will be finding AR that is radial component so the basic process is see Vector p Vector P doir Vector P doir the AR component that is Vector p doir means the component which is there in the direction of ir that is a r right so what is Vector P Vector P that is ax iix plus a y i y plus a z i z and dot ir is there dot ir is there so now separately we need to have Dot multiplication so ax into IX do I is there with us plus a y into iy dot I is there with us plus AZ into i z dot ir is there with us right now based on angle based on angle we need to identify this Dot multiplication so here first of all we need to identify angle of X and R so Here If You observe see this is x-axis and this is the axis of radial Direction means direction of R so angle between X and R that is five over here right so let me note it down here angle is pi If You observe this is what y- axis and this is radial Direction so angle between Y and Radial direction that is if this angle is y then this angle will be 90 - 5 so here angle is 90 - 5 and angle between I and Zed so here we have I and we have i z angle between I and I Z that is 90° so here we have 90° angle right now based on these angles we can easily say this a that is ax into as per cos we will be having calculation the reason is Dot multiplication explains what Dot multiplication explains cosine component so angle is 5 so IX doir that is cosy here we have a y into iy. I and angle is 90 - 5 so COS of 90 - 5 that is sin 5 and here we have angle 90 so cos 90 is zero so you don't need to write so a that is there with us ax cos 5 plus a y sin 5 right so now we got the value of AR now let us find the value of AI to identify the value of AI we need to follow same steps right see A5 is how much A5 that is Vector P dot I5 what is Vector P Vector p is ax IX plus a y i y plus a z i z and do I5 that we are deal with to do right now here separately we will be doing Dot multiplication so we'll be having ax into IX do I5 plus a Y into iy do I5 plus AZ into i z do I5 now we need to identify angle between X and 5 angle between Y and 5 and angle between Z and 5 so here we have five and here we have X so to understand angle let me take a trajectory so here we have a trajectory this is the direction of X we need to identify this angle right we need to identify this angle that is angle between pi and X see here we have angle 5 this is 90° in total angle is 180 of this triangle so what is this angle this angle that has to be 90 - 5 right this angle that has to be 90- 5 and see here this much angle that is 180 so what is this angle that will be 180 minus this much angle that is 90 minus 5 so 180 minus 90 that is 90-- plus 5 so angle between 5 and xaxis that is 90 + 5 so let me note it down here we have angle that is 90 + 5 that is angle between X and 5 now we need to identify angle between Y and 5 so y AIS is this and five is there in this direction so you need to take a trajectory of five so that is happening somewh somewhat like this you can observe it is happening somewhat like this now if you carefully observe this trajectory right so this angle this angle that is exterior angle of this right this angle that is exterior angle of this means if this angle is five then this angle has to be five only so angle between 5 AIS and Y AIS that is five so here we have angle that is five and angle between Z and 5 so you see five is there in this direction i z is there in this direction this angle is 90 so this angle is 90 right now here we have ax into COS of 90 + 5 that we will be having cos of 90 + 5 that is minus sin so here we will be having - a x sin 5 here we have angle 5 so we will be having a y cos 5 and here we have 90 angle so cos 90 is 0 so this will be zero and AZ that is equal in both so you don't need to convert AZ AZ is same for cartisian vector and cylindrical Vector so AR calculation that is as per ax cos 5 a y sin 5 and A5 calculation that is as per minus ax sin 5 plus a y cos Pi now we will solve one problem over here so to understand problem here we have been given with cartisian Vector you can observe and we need to convert that into cylindrical Vector right so to convert this in cylindrical Vector form first of all you need to have angle F based on angle F only we can convert this so what is angle five see angle 5 that is I have explained that in my earlier videos angle 5 that is tan inverse y by X right so angle 5 that is tan inverse y by X here we have X that is 2K3 and Y that is 2 so 2 / 2 < tk3 so we have tan inverse 1 by < tk3 and 1 by < tk3 tan inverse is how much it is 30° so now we have angle 5 that is 30° right now we need to calculate a first so what is ar ar is X component into cos 5 x component is 2 < tk3 so a is a X cos 5 + a y sin 5 and here a x that is 2 < tk3 ax is 2 < tk3 Y is 30° so cos 30 a y is 2 so 2 into sin 30 so 2 < tk3 cos 30 what is cos 30 cos 30 is < tk3 by2 and and here we have sin 30 what is sin 30 that is half so this 2 is getting canceled root3 into root3 that is 3 and this two is getting cancel so 3 + 1 that is 4 so we have AR that is four right now we need to identify AI over here AI calculation that I have explained what is AI that is - a x sin 5 plus a y cos y here we have all the values we just need to substitute that what is a x that is 2 < tk3 so minus 2 < tk3 sin 30 + a y that is 2 2 cos 30 so - 2 < tk3 sin 30 is half plus 2 into cos 30 cos 30 is < tk3 by2 so here this two is getting cancel this two is getting cancel minus < tk3 + root3 that will be 0o so A5 is 0 and one should know AZ that is as it is that is four only right so now we have all the coordinates we can represent this Vector into cylindrical Vector now so now our cylindrical Vector is Vector F that is a r that is four so a r into I Direction plus AI into I5 Direction plus a z into i z Direction here a r that is four so 4 I A5 that is zero and a z that is 4 so + 4 i z that is how I can represent this Vector in form of cylindrical Vector right so that is how basic process is there for a conversion and you need to remember these two equations for cylindrical Vector these two equations are very essential a r that is ax cos 5 + a y sin 5 and AI that is minus ax sin 5 plus a y cos 5 and one should know the basic process and if you do this practice one or two times then definitely you can easily solve this in examination but remember how to follow the say right thank you so much for watching this video still if anything that like to share just note it down in comment section I'll be happy to help you thank you so much for watching this video