 Myself Rohini Melgu talking about vector transformation. This is vector transformation part 1. Here we will be transforming the vector from one coordinate system to another coordinate system. In part 1 I will be discussing the vector transformation from Cartesian to cylindrical. Learning outcomes at the end of this video you will be able to transform the vector defined in Cartesian coordinate system to the vector in cylindrical coordinate system and vice versa. So, relation between Cartesian and cylindrical coordinate systems. Before we move to the actual coordinate system convergence, the vector transformation. Let us just recall the relation. What is the relation between Cartesian and cylindrical coordinates? The relation is like this. The cylindrical to Cartesian. If I know the cylindrical coordinates which are rho phi z, if I want to convert it to Cartesian, the relation is like this. x is equal to rho cos phi, y is equal to rho sin phi, z is same as z. And if I know the coordinates of Cartesian like x, y, z, how I can find the cylindrical? That is rho is equal to under root of x square plus y square. As you can see, if I square and add these two, I will get the value of rho. And if I divide y with x, I get tan phi and phi is tan inverse of y, y, x. z is remaining same. Now, let us look for vector transformation Cartesian to cylindrical. What does it mean? It means that I have some given vector A bar which is A x, A x bar plus A y, A y bar plus A z, A z bar. Since bar is nothing but the unit vectors fine, what this capital means? This capital means the values like the vector if I say 3 A x bar, 2 A y bar and 5 A z bar. That 3, 2, 5 are this capital values A x, capital A y, capital A z. If I want to transform this vector in the vector of this form, A bar is A rho, A rho bar plus A phi, A phi bar plus A z, A z bar. This is my required vector. Then how I can do? So, how I can proceed? So, when I am finding out for A rho, I should multiply this given vector, this given vector with the unit vector A rho bar. Then I can put this given vector which is A x, A rho, A phi, I need to find out. Out of that I am first finding out A rho which is A bar dot A rho bar. And A rho bar I can write like this which is the given vector A x, A x bar plus A y, A y bar plus A z, A z bar dot product with A rho bar. Now, when I simplify this or I simplify this bracket and I can write A x, A x dot A rho, A y, A y dot A rho plus A z, A z dot A rho. I know A x dot A x is 1, I know A x dot A y is 0, I know A x dot A z is 0. But I do not know the dot product of the unit vectors in different coordinate systems. I do not know the dot product of A x with A rho. So, for that we will follow this table. This is the table of dot product of unit vectors in cylindrical and Cartesian coordinate systems. So, this table is on one side you can see A x, A y, A z. On other side you can see the unit vectors in cylindrical A rho, A phi, A z. If I want the dot product of A x with A rho, A x dot A rho is cos phi. If I want A x dot A phi, it is minus sin phi likewise I can take. So, let us take this equation A rho is A x, A x dot A rho plus A y, A y dot A rho plus A z, A z dot A rho. This is the previous slide equation I have written as it, as it is. Now, let us find out what is A x dot A rho in the slide. So, A x dot A rho is nothing but cos phi, isn't it? A x dot A rho is cos phi. So, I can put it here. Then I want to find out A y dot A rho. So, A y dot A rho I can locate here. A y dot A rho is sin phi I can put it here. Similarly, I want A z dot A rho. So, if I locate it is giving me the value 0 I can put. So, I got the A rho which is the part of my result vector is A x cos phi plus A y sin phi. And once I know these values of phi and A x A y I can find out actual value of this A rho. I will show that solution of one problem as well in this video. Now, once I have calculated A rho let us find out A phi. A phi is A bar dot A phi. So, again I put this what is A bar? A bar is the given unit vector dot product with the unit vector A phi bar. If I simplify I get this equation. Now, my task is to find out this A x dot A phi A y dot A phi and A z dot A phi. So, refer this A phi column. In this A phi column take the values minus sin phi cos phi and 0 depending on what is written out here. So, when I put these values I get the equation of A phi as minus A x sin phi plus A y cos phi. No need to remember these equations you have to follow the procedure just remember that table 1. And A z is as it is A z is not going to change because z axis in Cartesian as well as cylindrical means same. Now, if I want to find this vector transformation I need to find out this A x A y A z. Now, I am doing reverse this is cylindrical to Cartesian given to me is a cylindrical coordinate system and I am trying to find out in Cartesian. So, this is my required vector unknowns are A x A y A z. So, A x can be written as A bar dot A x bar on both side there should be A x ok. So, A x is how I can do is I can proceed further like this A x is equal to the given vector dot product with the unit vector A x bar. I can simplify again I do not know the dot product directly of the different coordinate systems. So, I should refer table 1 again. So, I refer the table 1 and then the same equation I have repeated here. I can find this dot product A rho dot A x I can locate A rho dot A x yes A rho dot A x I have located it is cos phi I substitute it here. Then I can find out the A phi dot A x I can locate A phi dot A x in the table A phi dot A x is this one minus sin phi. So, I can substitute it here similarly I can locate A z dot A x which will be A z dot A x is this term and I can get this equation as A x is A rho cos phi minus A phi sin phi. Similarly, I can find out A y so when I am trying to find out for A y A bar dot A y. So, dot product with A y of the given vector now given vector is in cylindrical. So, I can simplify A rho dot A y I can refer this rule A rho dot A y is sin phi. No, then A phi dot A y is A phi dot A y is cos phi A z dot A phi is 0. So, I get A phi as A rho sin phi plus A phi cos phi and A z is as it is. So, likewise I can again prove it here like in the A z column if you refer A z dot A z is 1 with others it is showing 0. So, A z is as it is. So, vector transformation an example is shown to understand how to work out with this. Transform given vector A bar which is 2 A x bar plus 3 A y plus 5 A z at 0.235 2 cylindrical coordinate system. So, A rho is A bar dot A rho I need to find out in cylindrical. So, I should find out A rho A phi and A z. So, A rho is given vector dot A rho given vector is this one right dot product with A rho. Then refer the table one for getting this dot products. So, I can find out these values and I can put it here cos phi sin phi and 0. Now, this cos phi sin phi I want to substitute these values from where I can get some point is given to me as 235. From this point and by knowing the equations I can find out phi as tan inverse y by x in my point the y is 3 and x is 2. So, it comes to be 56.3. I can substitute this value in this above equation and I can put this values and I get A rho as 3.6. 3.6 and then I can find out A phi as well A bar dot A phi again I put given vector dot A phi I can simplify the equation. I can get the values of this A x dot A phi A y dot A phi and so on from this A phi column like minus sin phi cos phi and 0. Again phi already I have calculated as 56.3 I can put it here I can calculate its value I am getting 6.25 into tan inverse 2 minus 4. And A z is as it is A z in the equation is 5 so A z is 5. So, finally I can say that I have transformed the vector A bar which is 2 A x bar plus 3 A y bar plus 5 A z bar as equal to 3.6 A rho bar plus 6.25 into tan inverse 2 minus 4 A phi bar plus 5. So, this is how the vector transformation works. These are the references used for preparing this video. Thank you.