00:01
Hello students, for the first bit to represent vector a, 2z cap i plus 2x cap j minus 3y square k cap.
00:15
In cylindrical coordinates, we need to express it as a vector is equal to arer cap plus a theta e theta cap minus az ez cap.
00:32
In cylindrical coordinates, the units are er cap equal to radial unit vector and e theta cap equal to azimuthal unit vector and ez cap equal to vertical unit vector.
01:03
To find the components ar, a theta and az, we need to project a vector onto each of these unit vectors.
01:14
The projection of a vector onto er is given by a vector er dot er where dot product a vector dot er cap.
01:25
Similarly, the projection of a vector onto e theta cap is given by a vector dot product e theta cap.
01:34
Let's calculate a vector dot product er cap that is 2z cap i plus 2x cap z minus 3y square k cap dot product er cap.
01:55
Since er cap only has a radial component, we need to find the radial component of a vector which is in the er vector cap direction.
02:06
From the expression for a vector, we see that only term with a radial component is 2z cap.
02:14
So, the projection became a vector dot product er cap equal to 2z cap dot i that is equal to 0.
02:24
Therefore, ar equal to 0.
02:28
Next, calculate a vector dot product e theta cap that is 2z cap i plus 2x cap z minus 3y square k cap dot e theta cap.
02:44
Since e theta cap only has a azimuthal component, we need to find the azimuthal component of a vector which is in the e theta cap direction.
02:53
From the expression of a vector, we see that only term 2x with an azimuthal component is 2x cap z.
03:02
So, the projection is a vector dot product e theta cap is equal to 2x cap dot z that is equal to 0.
03:12
So, a theta equal to 0.
03:19
Finally, we will find a vector dot ez cap that is 2z cap i plus 2x cap j minus 3y square k cap dot ez cap.
03:36
Here we will get a vector dot e cap, ez cap equal to minus 3y square k cap dot ez cap, ez cap is equal to minus 3y square...